Powering Legacy Audio Speaker System

If you run your hdmi directly into your tv and optical to your reciever I would reccomend one of the older powerhouses like the Denon 5803. It's gonna sound better than any of the new light weight recievers anywhere near its used cost. Weighs 60lbs and is 170 watts a channel. Was over 4k new. 
Should be able to get a good one for under 400 if you look around. I lucked out and found a near perfect one in a Pawn shop for I believe $130. I also have a newer Denon 3808ci which was a $1600 receiver. The 5803 sounds much better. 

I'm new to this so I'm a little unfamiliar with the difference between receiver and integrated amp.  

A receiver usually is an all-in-one unit.  It usually refers to something that has an amplifier(s), preamp, tuner, DAC, and controls video.  It will decode surround sound input like DTS, Atmos, etc.  It may also have a phono stage.  They are generally speaking, a swiss army knife built more for home theater than high quality audio.

An integrated amplifier typically has an amplifier and preamp but no video or surround sound capabilities.  It may have a phono stage and some have DACs in them.  Integrated amps are typically 2 channel and purpose built for listening to music.  They may have a home theater bypass function so that it can work in conjunction with your home theater setup.