I use a 3.1 setup in my family room. A set of left/right main speakers and a center channel. I didn't have a good location for any surround speakers, so I skipped them. Put a good 10" subwoofer behind my couch (ran wires under the room) and the sound is quite an upgrade over the TV speakers.
I don't think you should need much more than a $500-$800 or so A/V receiver. That will leave a good deal of money for the other speakers and a good compact subwoofer. If you want more bass output, spending more on a better quality subwoofer will certainly give you the bass response you are looking for.
If you can put together a drawing of the room, or a few photos we can all look at, we can make some better recommendations on layout and reasonable expectations.
Wireless is fine, but the receiver shouldn't be your main focus for that. You can add a quality external wireless receiver if you need it in the future. But, many receivers today have Alexa/Google built in. So, Yamaha and Denon remain top-shelf choices for those types of features.