hykerupmypants
First , as I like tents (keep in mind that I look after Tarptent here in Australia, so more than a biased opinion here...)

"How much better is a $300 tent or sleeping bag than my $20 ones"
Not 15 times better. However there are many differences to justify the reason why a lot of keen hikers will spend that and more on a tent.
(or a squadron of tents in my case)
My fav (design wise because I have only used it in the backyard) solo right now is the TT Moment.
This is why : for a start it packs very small (about 20x4") , it is light (about 29 oz) inc pole and pegs.
It has a dry set up (the inside is sealed as you set it up) it sets up fast and has a lot of usable space inside (for a solo)
Now a small light tent helps towards not having to get a heavier larger pack.

So something like the Vapour Trail will likely do . I quickly tested that and the Gregory , both felt good but I have a very standard size back.
Try the pack on with the expected weight you need to carry plus about 4-6 lbs (extra water and or food for longer sections)

I had a quick search on the Golite quilt, looks that you can get that for $135, that will keep you warm from about 25f up (not 20...)
Also small pack size and light too.
That Campmor bag also looks the part .
(I am more into quilts now but whatever you like)

Those 3 items (TT Moment, Vapour Trail and Golite quilt) will give you for around your budget a DELUXE kit, not a starter one.
Franco

The Contrail works well too particularly if you use trekking poles. However you need to practice a bit with this one and learn how to use it. The Moment is almost difficult not to set it up correctly...