There is actually much more than just 3 pieces of evidence - BUT the 3 most basic are the following (note the following descriptions whilst NOT 100% technically correct (I'm ignoring relitavistic effects and much of the detail should give you the jist of the evidience)
1) as discovered by Edwin Hubble every distant galaxy appears to be moving away from us - and the further away it is the faster it is moving. This implies that a long time ago everything was closer together and even further back in history it must have all come from a single place.
2) the composition of the universe as measured many times over the past 50 years is roughly 75% Hydrogen and 24% helium along with 1% of the more interesting stuff (like planets and heavy elements) this is precisely what the big bang model predicts.
3) the Cosmic Microwave Background Radiation (CMBR) was discovered in the middle of last century - its a kind of afterglow in space from the big bang - and once again is precicely what would be expected to be seen.
4) radiation