What you want is gear that is designed for backpacking. Go on REI, look up all the items you are interested in and then hit them up when they have one of their many yearly sales and/or scout Facebook Marketplace, craigslist etc for the same gear in used condition. Their website lists the use case (camping, backpacking hiking, mountaineering etc) in the spec sheet and the reviews can be helpful.

I am a big backpacker and kitted myself out this way. All of the same gear translates directly to bicycle touring as well.

I understand being frugal, but a frugal minded person should also want the best bang for buck and should expect their gear to last a very long time so that they aren't buying again when they could have avoided it. For tent, sleeping pad and sleeping bag I would spend the most money and also probably not buy used. You will thank yourself down the road.

Lastly, the REI brand is not bad in many cases. Some of their gear holds up extremely well, some of it is junk (and getting worse as the years go on). For your needs I think an REI down sleeping bag would be perfect and not too costly. Same with their tents.