Anytime the pawn comes to c3 early, that means the Knight won't be coming there soon. Which means that the center squares it normally covers (e4 and d5) are weaker. Which points to Black playing Nf6 or d5 early to equalize.
Against 1..e5, the opening 1.e4 e5 2.Nf3 Nc6 3.c3 is called the Ponziani. It's not highly regarded as Black gets a fine game after 3...d5. 2.c3 looks like a weaker version of this, when d5 or Nf6 look great for Black. From a historical perspective, if 1.e4 e5 2.c3 were halfway reasonable, it would have been played a lot more frequently than it has. Of course you can play to gambit a pawn with the Danish Gambit: 1.e4 e5 2.d4 exd4 3.c3 dxc3 4.Nxc3, but I don't think this is what you have in mind.
Against the Sicilian (1...c5) , 2.c3 is fine. I know it only as the c3 Sicilian but somebody has probably named it somewhere. One of the differences here is after 2...Nf6 the pawn can advance to e5 with a tempo on the Knight.
You can't play 2.c3 vs. the Scandavian (1...d5) or the Alekhine (1...Nf6) because it loses a pawn.
Against other openings like the Caro-Kann (1...c6), French (1...e6), Modern (1...g6), or Nimzowitch (1..Nc6), 2.c3 can transpose but mainly it just limits White's options. It's like showing your cards in poker. No benefit and possibly a big drawback.
If you're wanting to play with the c3 setup consistently, try 1.d4 and 2.c3. Then you decide whether to play the black B outside the pawn chain before playing e3, or keep it home. There are various names for these systems (London, Colle, Stonewall, Torre Attack).