What is the best table top role playing game for beginners that can be picked up cheep???
Seconding D&D 5e. While not my favorite its probably the easiest to get ahold of. D&D 4e is a great game that gets a lot of hate because fan boys and memes. 4e's three core books Player's Handbook, Monster Manual and Dungeon Master's Guide you can get on amazon and ebay for like 10-15 each.
@Damyel I'd also like to throw out the Cypher System. Its a genreless system that can be used for anything. Fantasy, scifi, modern thrillers etc. The main books is called the Cypher System Core Rule Book and is everything you need to play. There are books that give you the core Cypher rules but tweaked for a unique setting like Numenera and Gods Fall.
Savage Worlds: Explorers Edition has almost everything you could ever need in a single book for less than thirty quid.