It's not DIY, but a Sansa Clip+ has good sound, goes pretty loud, drives most phones, is a nice thing to have, and is nearly inside your budget. You need access to a computer to get tracks onto it, but you must have that if you can post here. It's probably the single best value-for-money hi-fi buy.
A Behringer UCA202 USB DAC/ADC is another cheap, good-value hi-fi item, drives most phones, maybe a bit quiet with some, works for recording too.
You can get a DAC kit pretty cheap on ebay if you want something to build, or this looks like a pretty good headphone amp:-
...only U$22 shipped. I haven't built one though, so that's a recommendation on sight only. Needs a power supply, batteries will do.
Watch out with the DAC kits, you usually need transformers to complete them.
These last 2 ebay items probably won't make that big a difference to the sound you get, but they provide a bit of entertainment to build.