It sounds like there's been some changes which you can control, and some which you can't.
Three is hard for some kids. They sail through two and then whammo! three is a whole other story.
If it were me, I'd drop everything that isn't essential. He's in care five days a week-- I'm not saying this to make you feel bad, but that's a lot for a three year old. I'd listen to what he's saying about his extras and see what's beneath that. Don't want to be nice in karate or sports? (which is not essential--he's three)-- stop taking him. That's just more time he spends being told what to do instead of getting to just play with you. Consider that he's three and along with having to behave and control his impulses during preschool,he also now has to do that EVEN MORE. Instead, I'd take him home and offer him some down time, some unstructured time. Go to the park and let him play with you. Do enforce your expectations of being polite, and also remember that at 3, he is still 'learning' about other kids and their feelings. It's very common to hear young kids (and even older ones) say "I don't want to play with you".. He may get PLENTY of time with other kids at school already; maybe when he goes to the park, he wants to be left alone to play (and then, teach him "no thanks, I play by myself) or might want to play with you.
Charts and rewards will only go so far. Threes aren't really able to reflect on their actions in the ways we are, or to assess and then modify them to achieve a goal. (There are a few, but from what I've seen in my long years of working with kids, it can be rare. Incentivizing one specific skill-- like using the potty-- is actually easier for most kids than using incentives to change behavior.) Here are some things I would do if I were in your situation:
Offer kind correction at every turn. When he's rude to someone, offer rephrasing (Like at Grandma's: "I'm happy to see you and I'm sad Mom has to go".) If a child is sad by what he says, draw his attention to that ("Do you see his face? He looks very sad. He wanted to play and you used a scary/mad voice. Let's check in with him. 'Are you okay?' ... etc.)
Limit the 'extras' for now. Your child may be really just wanting some unstructured playtime for now. Remember, little kids really don't need more classes, they need more of us and more unstructured playtime, which they will learn SO MUCH from.
I would not take the iPad or other toys away for his expressing himself in the way he is. Instead, offer corrections in the moment; rephrasing or authoritatively deciding "I see you aren't feeling friendly; we're going to go (for a walk over here, just the two of us/go home)"... no asking.
Tantrums? He needs to move through those emotions, so give him a safe place to do it and let him alone. Once we have tried identifying/correcting the source of their frustration and see that it isn't helping, it's often better to just let him have a safe place to rage and then reconnect afterward, when he comes out and is ready for company. I never try reasoning with a child during a tantrum. That's the absolute wrong time to give attention.
Lastly, give as much positive attention as you can. Praise the actions you want to see more of. I will suggest JoAnne Nordling's "Taking Charge: Caring Discipline which works at home and at school" as a great guide for the most effective ways of disciplining and giving attention at the most advantageous times. Especially when your son is busy and enjoying something he likes doing: this is the very best time to go over and give him a squeeze on the shoulder, ruffle his hair, kiss his head, rub his back-- it shows him that he is loved for *just being* (a very real 'step' on Mazlov's Heirarchy of Needs which must be fulfilled for the child to succeed.) This practice is called "positive attention during neutral times"--this is when they are doing something which doesn't bother us or is not meant to please us. I do swear by this method of giving them attention-- it does change behavior and you have to do it a LOT -- we aim for at least 10+ times a day.
MartyMomma was right--your son is watching, because he wants your attention. Be strategic about how you spend your time together (at three, doing less is actually more) and keep clear on your boundaries.Good luck!