Two of the bigger differences between P101 and W101 are the battle mechanics and the pet training. The questing and humor are pretty similar.
The battle mechanics will take some getting used to, but's it's still a matter of setting up your deck to get the cards you need when they matter the most.
Pet training as compared to P101 is an energy sapping, time-consuming, ordeal that you will either love or hate. You actually need to run your pet through their paces on one of five mini games that sap your green energy. As opposed to sending your pet off to school.
As to actual advice for the game. You didn't say which school you chose, so I will offer this. Do NOT spend all your training points on a secondary school. You'll want to train in Death up to Feint, Ice up to Tower Shield, and Life up to Satyr (don't worry if you do the Zeke quests through each world, you'll have more than enough training points.)
Side Quests are beneficial so do not neglect side quests. Actually at higher levels you miss out on certain spells and pets if you have neglected side quests.
But most importantly, if you have ANY questions, ASK. The denizens of the message boards are ALWAYS glad to help.
Welcome to the Wizard101 forums. For me, the big difference between Wizard101 and Pirate101 was the "independence." Because you're able to control companions in Pirate101, the game is pretty soloable (minus a few tough dungeons here and there). Although Wizard101 is definitely soloable when you get the hang of things, expect to need to ask for help more often than you do in Pirate101. I think the game is much more team-oriented in that sense.
Aside from that, I agree with everything Area51Alien said. :)
Just to add to the list, another difference you may notice is the game updates.
Wizard101 seems to undergo major updates (where new worlds, spells etc. are added) at least twice a year. As you know, this is not the same case for Pirates (actually temporarily suspended my P101 membership because of this).
So, you can therefor expect to have quite a lot to do at any given time while you quest here.
The Drop rate is worse than how it is on Pirates. Not sure of your experiences with farming over on Pirates, but hunting for gear from bosses can be very tiresome. Even simply defeat and collect quests from mobs can be lengthily too.
Most things remain for the most part pretty similar though, such as the questing, and moving about (only thing, we don't use ships here :D).
Wizards has fishing for cool looking fish and treasure chests and also daily and weekly fish to catch for rewards(extra gold); gardening with some funny looking well designed plants; crafting--clothing, gear, mount; and searching for reagents(craft items) while your wizards is traveling. Daily quests with rewards for crowns, gold, or treasure. Many things to do. And like Pirates--a great story line.