You only need HDMI connections on your receiver if you need you receiver to switch the video signal. If you run HDMI directly to your TV, then you may need to switch inputs on your TV when switching between sources. You would also need to run an audio signal to your receiver, since HDMI carries audio and video.
For $500, I think you have lots of good options for receivers. I like Pioneers and a lot of people seem to be happy with Denon and Yamaha. I'd just do some research on reviews and specs for some receivers that meet your needs.