On that budget you will be looking for vinyl roll up boards and plastic pieces. You should match the square size of the board to the pieces. A common recommendation is 2.25 inch squares and 3.75 inch King height, although some 4 inch Kings would probably work too.
Unweighted plastic chessmen are very light and should suit you fine. These basic chess pieces cost $6.49:
http://www.wholesalechess.com/shop/chess-pieces/club-chess-pieces/basic-club-chess-pieces
Here is a 20 inch vinyl board with 2.25 inch squares for $6.99:
http://www.wholesalechess.com/shop/chess-boards/vinyl-rollup-chess-boards/quality-vinyl-chess-board
A mousepad board will lay flat easier but cannot be rolled up for storage:
http://www.wholesalechess.com/shop/chess-boards/vinyl-rollup-chess-boards/mousepad-chess-board
I would also recommend getting a chess clock because they are a lot of fun to play with. A really good one wont fit your budget, but you could easily afford a cheap analog one (I actually like analog clocks a lot, but they are only useful for home games these days), good inexpensive digital clocks are the DGT North American or the ZMF clocks.
You can also download the chess.com chessclock app (should be free) for your iPad if you want to save money. It has all the features of the better digital clocks, but is not as much fun and cannot be used in tournaments.
I'm new to chess, doing iPad tactic exercises and watching videos. My 10 year old nephew is also learning, and we finally played a game on my iPad. I'd like to get a board so we could play easier, and I've read that 20" is the tournament size. What would people recommend as far as a board/pieces? I'm disabled and have problems with motor/dexterity, so anything with heavy pieces won't work. I'm on a budget so I'd say $50 is my absolute max budget, but less expensive is better.
Thank you for any feedback!