First time home theater for a complete noob - need help!!!

I just bought myself a 65" Panny VT50 and am currently remodeling the room I'll have it in. I'm finally going to have room for a 5.1/7.1 set-up though I'm also condering a soundbar instead. I'm really not sure how all this works and my budget is $400 max, though I'd like it to be as close to $300 as possible. I've always just used the TV speakers (Yea yea yea ) so this will be my first high-quality sound set-up.

I'd like to keep this as simple and as user-friendly as possible and I do have a handful of questions for you guys:

Can the speakers just be run out of the TV so they work just like the built-in speakers or do I have to have everything set through a receiver? I mean, to get sound for each device I use (cable box, game system, PC, etc) can I just change inputs as I do now via the TV or would I have to change inputs via the reciever? If that even makes sense

The less wires the better, so ideally, I was wondering if there are any 5.1 systems with wireless rear-speakers to avoid having to run wires throughout the room?

Lastly, what would you guys recommend for my current price-range? They don't have to be some super-kick-ass speakers, just decent is fine with me haha.

Thanks!

Thank you for the reply! I've been able to up my budget to $500 (MAX) and I found a factory refurbed Onkyo S7500 for $499 <- Would that be the best route for me to take?
Also, I fully understand that the speakers still need to be plugged in - not a problem as there's actually outlets on either side to where I plan on putting them. The issue would come from running wires to the receiver.

You are correct, I realized this after I made the post. Apparently there's only one or two HTiB's which come with actual wireless speakers and they are fairly crappy sony and samsung models which is sad. I suppose I could always run the wires through my ceiling tiles so it may not be of issue.