After 2.Bc4, it's better to play Nf6 immediately rather than Nc6. It covers h5, prevents Qh5, and forestalls what subsequently occurred to you. It is a bit embarrassing, I agree, but like LisaV said it's just a trap.
Look, you haven't been playing seriously in a long time, or against even reasonably strong opponents, so it's really not a very big deal to make a mistake like that. It's a silly oversight.
Play long (time control, I mean) games for the moment and follow the rule of looking at all possible captures your opponent can make after your move as well as all checks. You probably won't be mated in less than 10 moves if you just take your time and do those two things. Attacks on f7 are common after 1e4 e5 and the king's pawn openings can be very tricky.
Maybe play 1.d4 as white, and the Caro-Kann against 1.e4, as black, to try to avoid some of the immediate sharpness that can go on after 1.e4 e5.
Try some correspondence games, too, where you can take your time and analyze what to do.
Finding someone equal, or even better, a bit stronger than you to play OTB will help a lot. Most of my play now is OTB, which is why I only have one or two games on this site even after being on for a year.
Most of all, forget about the embarrassment of losing. It's going to happen. You win some and you lose some.
Last weekend I dropped a rook outright to a far weaker player after having gone up a pawn in the opening by what I thought was reasonably good play. I went on to win back the rook, but in reality I was clearly lost after his coup. I was able to temporarily trap his queen, though not to win it, and then gain some play with my own queen and a couple of pieces. Blunders happen. You just have to try to minimize the number of times you make them.
Ok, I watch a few videos, buy a book, try to remember some of what I learned in HS/College (I am in my 50s, so that is a while back). I play against the Mac free computer game for practice.
Finally last night I go online, accept a challenge from someone with about a 1400 rating. I was checkmated in 4 moves. Now I had a bit of a headache -- but still it was embarrassing. Now I am a little gun shy about trying again.
I keep practicing again the computer. Then I watch a video or do a puzzle, etc. I feel like I am learning and some things are coming back. But chess, as I am finding out, is NOT like riding a bike. You forget basics, you have to learn them again.
My son is a HS Senior and he and his friends play and invite me to join them every so often. I do have a Ph.D., so I wallop them -- but they do not study or learn opening and endings, etc. They just play the board.
If I got checkmated again in under 10 moves I would be TOTALLY embarrassed. I know the only way to get better is to play. Also the person who beat me logged off and immediately jumped to another game. I was hoping for a bit of conversation -- at least telling me I missed the obvious would have been nice.