When May starts her journey at 10, I imagine Dawn is about 8 at this point. Before May began traveling, Drew had already been competing in contests. Dawn, who knew her goal from a young age, excitedly would watch contests whenever she could. She began seeing Drew in contests and became eager over him as he was a brand new coordinator that was doing unusually well for a beginner.
She developed a chidlish crush due to her admiration (and it didn't hurt he happened to be pretty good looking to add to talented), always rooting for him in his contests. By the time she was old enough to travel, she was over her silly crush, but she still admired Drew as a coordinator.
She had seen May in contests, including both of her Grand Festivals. Dawn didn't think of her much when May first started out, but she began liking her more as she grew and became stronger and began to admire her. To see May and Drew battle in the Kanto Grand Festival--and to see May win--really unexpectedly pumped Dawn up. For someone who started out as such a rookie to overcome someone like Drew was really inspiring for Dawn. She always hoped she could meet her, but wondered how likely something like that would be.
When Dawn found out that Ash used to travel with May, her mind was blown. Her chances to meet someone she admired had risen, but suddenly... she felt unsure if she wanted to meet May after all. Dawn was going through a slump, and wasn't anything special as a coordinator. How could she face someone as impressive as May?
May's visit was very unexpected. Without warning, May was in Snowpoint City about to head over to meet them so she could enter the Wallace Cup. Dawn became anxious in the day she waited for May--how could she compete knowing someone like May would be her competitor? There was no way she could win, let alone not make a fool of herself. May, on the other hand, was eager to visit. Ash and Brock had filled her in of Dawn's slump without Dawn's knowledge, and while May had never had Dawn's exact problem, it still resonated with her. This trip wasn't just about getting a ribbon--it was about Dawn.
Things end up really well. Dawn meshed very well with May despite her intimidation of her and ends up seeing May as an equal by the end, her win a huge confidence boost. But even after May leaves, she still sees May as an older menor-ish figure as well as a friend, always very open and eager to hear her advice and opinions when the two talk from time to time (and the two do continue to talk through letters and e-mails and stuff as well as often watching each other' contests if they happen to be by a TV in the Pokemon Center).
When Dawn is off on her own in Hoenn, she enters the Wallace Cup once more which is now being held in Johto. She's super excited this time around because this will be the first time she'll see May again in person, but this time around, Drew is there as well. Not only was Drew someone she had always admired, but now, he was someone she constantly heard about through May (which basically means she now knows all of his difficult and unbearable sides as well as knowing May totally has fallen for him and will tease her for it despite May's protests).
Drew has heard a lot about Dawn as well from May, and tells Dawn he watched her in the Wallace Cup and Sinnoh Grand Festival and was impressed. Dawn is flattered to hear this form someone she had admired since childhood, and May suggests the two have a battle before the Wallace Cup. Dawn freaks a bit, but Drew is polite and says it would be an honor to battle the runner up of the Grand Festival as well as a close friend of May's. The two battle, and Dawn loses, but she doesn't even seem to care, feeling as though she learned a lot from the experience as well as excited she could battle someone like Drew at all.
Later on, May explains to Dawn that Drew has been having a bit of trouble with Super Contests, which surprises Dawn. May says that Drew had managed to win a ribbon by now, but that it was still troubling him a bit. May had been helping him ever since she got Drew to admit he was having difficulties, and explained that this was the first time Drew was entering a contest after he had taken some time off to focus on his problem. Dawn can't believe how much she connects to Drew's situaion, and that what's most surprising is even a Top Coordinator like him still has obstacles to overcome just like she does and will continue to have.
Drew ends up winning, and Dawn feels really happy for him. She congratulates him afterwards, and Drew thanks her politely for her words. Dawn goes to the docks to see off May and Drew who are now currently in Sinnoh before she heads back to Hoenn herelf to get in a final good bye. Truthfully, Dawn had been feeling a bit lonely ever since she had gotten to Hoenn--but she's reminded that she has close friends and people who will support her that will always be with her, even if there's distance between them. She feels energized to return to Hoenn on her own with what she's learned from the Wallace Cup, and her friendship with May feels even deeper to her now--and she makes sure to tease May one last time about Drew before she boards the ship to meet him.
She certainly remembers the support she has later when Zoey calls her that night to compliment her performance--and then to critique it.
