Omega has great boar offerings for less than 20 bucks. Great brushes. Let us know what you plan on using and we can give you additional suggestions. My Tweezerman has served me just fine also.
If I am going to go less than $20 I'd probably find a good boar brush. Omega, Semogue have plenty in that range that many would argue are better than their badger counterparts. Maybe even the Vulfix 2006, a mix of boar and badger.
A little more coin gets you into a good grade of badger, the $20 to $50 dollar range. There are so many choices then. Simpsons has a few Pure badger in that range, Rooney Pure badger 3/1 at vintageblade is a nice one. Nicer Vulfix badgers. Even some from craftsmen available from the boards.
You asked for a 'good' first brush, doesn't have to be a 'cheap' first brush. If you have a price range in mind the guys here can help narrow down that search a bit. Also if you know if you are going to use creams or soaps or possibly both. Let us know what you got, I am sure people are going to give great suggestions.
Omega boar brushes are a great start and loved my many of the veterans here. Im a Semogue boar lover but many of the recommendations you'll get will be great.
If there is one suggestion I will give is to keep it simple because you may like the $20-$40 range brushes more than the top shelf $100+.
www.penworks.us has Omega line and great priced badgers in the Penworks line. Very reputable member here.
My first (and still only) brush was the Crabtree & Evelyn Best Badger brush, for about $35. I've never used anything other than it, so I can't offer comparisons, but I've been very happy with it.
Why not buy a not-too-expensive boar AND a not-too-expensive badger?
That way you can try what you like best. Or, find out you like the boar better with soaps and the badger with cremes. or the other way around.
In boar, I agree Omega and Semogue cannot be beaten. My 1305 boar was like 12.95, i remember. The cheapest badger is a DIY (see here, we just started). Or check Omega (Penworks is recommended indeed) for their pure badger.
If you are looking for something more affordable, a Vulfix boar. If you can spare a little coin, a Rooney in super. They cost about $80 shipped but if you are patient you'll be able to pick a used one up on the B/S/T forum for around $60. Excellent brushes.