I bought Dragonborn about 2-3 days ago, yet haven't been there yet. I was attacked by some Cultists outside of the Sleeping Giant Inn, but figured I should 'finish up' in Skyrim first.

I am about level 20-23 (I never pay much attention) and I have finished:
-The Companions
-Civil War (Empire)
-Main Quest

I am a warrior, heavy armor+block+one-handed. I wear mostly Blades gear and use the Blades Sword Dragonsbane, however I DO have Dawnbreaker and Ysgrammor's shield if I ever need it. I have about 10 shouts

Do you think I should do anything else before moving onto Solsthiem? I also have Dawnguard, yet haven't done anything with it either. I just didn't know if certain quests and events from Skyrim/Dawnguard would make my experience in Dragonborn better, or if I should be on a higher level and stuff.

i would go thru the dawnguard main quest,the set sail

Try to get to level 25.

Be sure to have a good amount of Dragon Souls, at least 15 for the new Shouts, you'll actually need three for the story.