I am thinking of buying a ps4 but on my ps3 I have brought a few games off the ps store. If I get the ps4 and sign in with the same account will i be able to re download the games?
Most PS3 games are incompatible with PS4. In fact, many PS3 games have their own PS4 Versions, such as Devil May Cry Definitive Edition and The Last of Us.
Games on your PS Store are there to be downloaded anywhere but will only work on compatible systems. You can find the version they work on at the game page on PSN. Some games (like Another World HD) have tags for both PS3 and PS4.
Edit: Alternatively, you could use PS Now to stream PS3 games: https://www.playstation.com/en-gb/explore/playstation-now/ps-now-games/