It really depends on your continuation. At GM level 2.Bc4 is sometimes played with the idea of reaching a KGD/vienna setup. In itself not very sharp, the plan is to build a slow but persistent attack on the kingside with a setup including e4 Bc4 d3 Nc3 f4 Nf3 0-0 in some order. A good source is Attacking with e4.
W can play more aggressively and build a gambit repertoire around the bishop opening. http://www.rci.rutgers.edu/~goeller/urusov/ presents a complete 1.e4 e5 repertoire based on 1.e4 e5 2.Bc4 Nf6 3.d4!?. this is probably good for no more than equality, but still a lot of fun in some variations.
You only have to be aware that the least theoretical way for an amateur to face 1...e5 is probably the spanish. This is because W setup is very strong and you can reach a position which is never worse than equal without knowing any theory. The bishop opening is not so pressing and as a consequence it needs more work to play confidently.B has tons of replies. And among the players i know noone has studyed as much theory as the one who plays the london system, because here B can really play almost everything. The more forcing your setup is, the less you have to study (this of course is only true until you reach a certain level)
So my advice is: if you play 2.Bc4 because you like the positions, welcome to the fascinating Bishop opening. If you just want to rattle out the opening moves without thinking, then chess is simply not the right game.
There is no telling black will actually play those variations. Black didn't play the first game well. White is fine after those odd queen moves. The sacrifice in the second game is speculative. You might have made it easier with 4.Be2. As it went, maybe black should have exchanged the bishops. I do have mixed feelings about Bc6. Taking another pawn on d5 costs a lot of time, but just standing there on c6 it looks vulnerable. You also activated the Rb8. On move 8...Qc7 looks interesting for sure. It sacks another pawn and it is probably a bit much, but white is lagging in development. It's exactly what you want as black with this speculative setup.
Instead, if black plays a ''normal'' opening and 3.Nf6 you are probably looking at an Italian or a Vienna game. On 3.d3 there is 3...c6 and on 3.Nc3 there is the wild 3...Nxe4 If black is not adventurous you can set up what you want, as said above.
