Since both Lost Mine of Phandelver and Dragon of Icespire Peak take place in the same region in the Forgotten Realms, it's often recommended to run them together. However, doing so requires (at least in my opinion) some alterations to one or both adventures.
LMOP includes a young green dragon Venomfang in the ruins of Thundertree, while DOIP implies the whole region (including Thundertree) is now the territory of the young white dragon Cryovain, whose lair is in Icespire Hold. At the very least, in my opinion, you need to rid yourself of one of the dragons for combining the modules to make sense.
Option 1: Remove Thundertree
This is probably the easiest of your options. Thundertree is a completely optional dungeon, especially if you're using some of the quests in DOIP to fill out LMOP, so the players will have no trouble obtaining the amount of experience they need to continue the adventure.
The players can either gain the druid's trust another way, or find the locations of Cragmaw Castle or Wave Echo Cave in another way than being guided by the druid.
This also allows you, if you want, to completely drop the plot with the Dendrar family if you wish.
Option 2: Rekey Thundertree and Remove Venomfang
This is the second easiest option in my opinion, as it doesn't involve significantly changing either module, but merely rekeying a side quest dungeon.
To do this, you'll need to remove both Venomfang and the dragon cultists in Thundertree and replace it with something else.
This is the option I've been running with (despite really liking some other more complicated options), and I've put in some different cultists, who are working for a potential down-the-line bad guy. They're purposely trying to keep Thundertree corrupted, and there is a corrupted treant in the area to replace Venomfang.
You could do anything with Thundertree though, especially if you're trying to seed the adventure after the current one for your party.
Option 3: Keep Venomfang as written, remove Cryovain
So this is where you'd have to start re-writing the campaigns. This option prioritizes maintaining the plot as written in LMOP, but rewriting significant portions of DOIP.
If you go this route, you need to know why the orcs are leaving their old territory for the area near Phandalin, especially given the goblinoid activity in the area. The implication in DOIP is that the orcs were living in Icespire Hold, but were pushed out when Cryovain arrived and made it his lair. you could keep this basic concept, but then you'd need to replace what took over Icespire Hold with something or someone else.
This isn't necessarily a bad idea, since as-written, Icespire Hold is a bit of a letdown of a dungeon, but it involves more writing on your part as a DM.
Option 4: Keep Venomfang as your dragon, and also she's behind everything
One of the big complaints about both DOIP and LMOP is that the final bosses are something of a let-down. The Black Spider doesn't get built up nearly enough in LMOP, and isn't a particularly challenging encounter, and while Cryovain can certainly hassle the party long before they arrive in Icespire Hold, the dragon makes up very little of the module.
This rewrite (which I would like to run someday), removes the Black Spider, and Venomfang becomes the mastermind behind the Redbrand/Cragmaw alliance, something quite in-keeping for the typical green dragon.
The orcs could still have been pushed out of Icespire Hold by an unrelated threat, or it could be by a related one, or they could still occupy Icespire Hold, and the Talos cultists we meet in DOIP may also be working for Venomfang.
This version is definitely a bit beyond me as a new DM, but it's probably my favorite resolution to this issue, and to the finales of both LMOP and DOIP being a bit underwhelming as written.
