Is playing poker on the Internet safe?

Depends on what you mean by 'safe'. Most players worry about losing their money that they deposit. This has happened before (look up "Pokerspot scam" if you want the details) in the earlier days of online poker, so it is a risk. That said however, the established poker sites are your safest bets, since they are regulated and also have their money (your money) backed by big financial institutions.