What does Hornet's training look like? Does she prefer to train alone or with company? Is there any part of training that she enjoys or dislikes the most?
You can apply that question to other characters as well!
I might as well go through her history of training/fighting since it's a good opportunity to talk about it while also answering your question. And since I'm mostly focusing on her training/sparring with a weapon, which may not be what you asked (but is the main way for her to train), I'll also mention some physical training later, to get a more complete picture
She first showed interest in fighting when she was in her early teens, watching the Great Knights of the Pale King training. She had that kind of itch since she was a pup, she went through (and sometimes chewed through) many nails made of shellwood, but that was the first time she really wanted to do it.
She asked Vyrm about it, and he promised to find her a mentor. In truth, he already had a plan that he discussed with her mother, Herrah, prior to her becoming a Dreamer. She was to train under the Hive queen, Vespa, with the Needle made for her from Hivesteel, that Vyrm stored away for this occasion.
And so he contacted Vespa and made a deal with her to arrange Hornet's training sessions. The Needle was sharpened after the years of waiting for its wielder, and given to Hornet on the day of her birthday. What Hornet did not know, was that Herrah and Vespa went back and were closer than she realized. Consequently, she did not see Vespa as anything else but her mentor, though looking back she definitely remembers Vespa's fond glances at her, that she didn't think of too much at the time.
The training sessions were definitely very demanding, there were moments Hornet was ready to throw the Needle aside and quit. She got angry at herself and at Vespa at times, though she pushed through. Her least favorite parts of the training were the boring theory and basics. Her most favorite, as you can guess, was the actual fighting part. It was so thrilling to her, the sound of the blades clashing, the energy boost of successfully breaking through the defenses and getting a hit, or dodging her opponent and confusing them. At first, she trained with the Hive Knights under Vespa's watch, but eventually she got to fight the queen herself, and she still remembers that to this day.
But then everything fell apart, her father vanished with the palace, the infection returned, and eventually Vespa fell under its grasp and passed away. With no home left, Hornet wandered. And with no home, there were no training sessions. Now her skills would be put to the test, and the real fight would begin. She was still young, still a teenager, just about to reach the age when most bugs reach their maturity. But her skill was apparent as she cut through the beasts that tried to kill her, or the poor infected bugs who lost their minds.
Her itch for one-to-one combat never left, however, and as she was forced to earn geo to get any kind of services from the few surviving bugs of Hallownest, things such as healing supplies (her silk might've been a lifesaver, but her wounds still needed cleaning and extra treatment) or any repairs to her tools and the Needle, she turned towards the Colosseum of Fools.
It was a far cry from the duels she fought with Vespa. It was dirty, unfair combat that would end in bloodshed, all to amuse the audience and generate profit. But it was the best thing she could've done to earn anything. And so she signed up under an alias, The Sentinel, all the while hiding her identity with clothing and facepaint. She wasn't a known figure in Hallownest, despite being the daughter of the king, she wasn't hailed by the masses as the princess of the kingdom. To all the bugs in the audience she was a nobody, but she did not want to connect her name, one of the few things she had left from her mother and father, to such barbaric combat.
She fought, she killed and she even became a champion for a while. However, she quit once she felt she had enough geo, and because she simply had enough. She was no stranger to killing, there was that voice in the back of her head to rip things apart, spill blood, to consume the flesh, and she had to actively suppress it. With each fight and more and more drops of blood from her opponents, and all the cheering from above, she found it more difficult to control herself. She had to stop, and she did. Her alias eventually faded from the memory of most bugs in the crowd that watched it, with only a few still remembering the glorious fights The Sentinel would emerge victorious from.
In the years after, the only fights she ever took part in were in self defense. That is, until vessels started showing up throughout Hallownest. They must have found a way to get out of the Abyss, and most of them were unstable or clueless. She knew enough about void to determine that void beings roaming around the kingdom were bad news, and so she eliminated the threat. They were her siblings, but she pushed that thought away. At the end of the day, they were just simple beings with no thoughts or feelings. That is what she told herself whenever guilt struck.
Eventually, a vessel appeared who would push her towards reexamining how she viewed them. Ghost of Hallownest, she called it. They fought well, and managed to beat her, and then continued on the journey. As it turned out, it was set on freeing Hallownest from the infection. She fought it again, but that wasn't enough to stop it. Eventually it disappeared, and soon after, the infection followed it. She has never seen that vessel again.
Her fighting history doesn't continue in any notable ways until after the return of Vyrm. Her anger issues that built up in her heart for all the years were unleashed, and as she berated and yelled at him, she slowly realized it was consuming her. His near death experience was the point where she decided it needed to stop, and she turned to Grimm for help.
Grimm offered sparring sessions as a way for her to work on her self control. He wielded no weapon, only his Nightmare Cloak, which proved to be an incredibly tough opponent, and a defense that was near impenetrable. What could've been frustrating, turned out to be a very effective way of Hornet tiring herself out until she had no energy to be angry. And in those moments, she was able to look more clearly at the world around her, consequently finding ways to maintain that clarity through a lot of self care work.
They still continue those sparring sessions, now less as therapy and more of a bonding experience. And with the arrival of Aika, Lace and Phantom, she also gained more sparring partners.
With Aika, Hornet gets to assume the position of a mentor. Aika's skill in archery is incredible, but she struggles in more direct combat. Hornet, meanwhile, is one of the most skilled blade fighters in all of Hallownest. Her main weapon is the Needle, but she also gained near mastery of smaller weapons like daggers, which seem to be what Aika gravitates towards. Their sparring sessions are quite clumsy compared to other sessions Hornet remembers, but they're very amusing for both of them, and at times tense in the way you can imagine for two bugs who are very attracted to each other.
Lace is another sparring partner, one who is equal in skill, but who mastered techniques Hornet has never seen before. Not to mention, where Hornet excels at her strength and endurance, Lace's speed and agility are nothing to joke about. Their sparring sessions are a joy to watch, two masters of very different techniques clashing as equals. With a lot of playful teasing, of course, and even more of that tension mentioned earlier. There are times when those sparring sessions become training sessions, however. Lace is a master of her pin, but she has little to no experience using other weapons or tools, so she has a lot to learn in that area, which Hornet is more than happy to help her with.
Phantom is the last potential sparring partner, but due to their disability, those sessions are much more gentle and act more as exercise for them to maintain the strength that they do have. To best describe their style, they're incredibly deadly, but in short bursts, after which they need a moment to take a breath. The longpin, which mainly serves as walking aid for them, is a weapon not to be underestimated, and neither is the skill Phantom possesses at wielding it. Hornet enjoys sparring with them a lot, it's a very different experience that she finds exciting.
To sum up, it's very clear that Hornet loves fighting and sparring. But now that she has the time to slow down, she also finds a lot of enjoyment in physical exercise. It's something she can actually bond over with Vyrm, who has gained a lot of muscle since settling down in Dirtmouth. They enjoy meeting and exercising together, and they both do it for very similar reasons - their bodies are naturally very durable, but they're built for hunting and surviving. The new peaceful life means they don't have as many opportunities to put their bodies through the test during hunting or fighting for territory, and so they have to make up for it at home. Weightlifting, endurance training, even occasional jogs around the town.
Hornet's favorite part is definitely the ability to spend some time with her father doing something they both enjoy. She also loves the energy boost she gets after a good session, though she's not a fan of how sweaty her body gets.
Vyrm is largely the same, he loves that he can bond with her daughter like this. And he also loves all the nice comments he hears from Grimm about his physique. The sweat is something he's especially irritated by, it makes him sticky and that's bad news whenever Grimm makes any physical contact with him (have you ever pet a cat with your hand sweaty from summer heat? Imagine that but 100x worse).
This response is already super long so I'll just mention other characters who occasionally join those "gym" sessions. Grimm does it even though he probably doesn't need to, but he can't say no to the sight of Vyrm. Hornet often shoo's him out of the room because all he does is ogle Vyrm and makes no effort to keep it subtle. Aika also joins them sometimes, she's quite fit herself and she likes to maintain that figure, especially now that she has two mates who very much enjoy the sight. As the kids get older, Asta would also join. Lewk wouldn't be the biggest fan of exercising, and Milo's health issues make it so he has to be very careful about exhausting himself, so he usually just watches. However, whenever he's not ogling his husband, I like to think Grimm would do some very light exercises with Milo so that he doesn't feel excluded (and because it would still be good for his health, just has to be done in moderation).