I'm married to an US citizen and I was wondering f I could wait for my green card process on American soil traveling with my ...

Your choice. It will be a faster process if you are in the US.

You should not enter the U.S. pretending to be a tourist, but with the intent to immigrate not just visit. Misrepresentation regarding your intentions when speaking with a U.S. Customs and Border Patrol agent is a serious problem and can get you barred from the U.S. Don't take chances with your family. If you want to consider your options, consult privately with a licensed U.S. immigration attorney.