How effective are condoms when it comes to preventing the transmission of HPV?
The limitation with condoms and HPV is the skin they don't cover. The virus won't penetrate a latex condom, but HPV can be on genital-area skin for which a condom won't be effective (base of penis, scrotum, perianal area, etc.). In one study with college-age women at the University of Washington (Winer et al.), consistent condom use by their male partners was associated with a 70% reduction in HPV acquistion for the women. Your mileage may vary, of course, but that gives an idea that while condoms certainly reduce the risk of HPV transmission, they cannot eliminate it.
How likely is it that I will contract HPV after having safe sex with him?
There's no way to put an exact figure to the likelihood. HPV is usually cleared by the immune system over time; most cases of HPV become undetectable on DNA tests within 6-24 months. This doesn't mean for sure the virus cannot be transmitted, but many experts believe the risks of transmission probably diminish over time (again, this remains unproven).
Keep in mind that HPV is incredibly common, almost all sexually active people are estimated to have one or more HPV infections in their lifetimes. Most are never diagnosed, don't result in visible symptoms, and are cleared/suppressed naturally by the immune response. If you're not doing so, make sure you're going for regular Pap tests. If you do become sexually active with this partner and he had genital warts, just make sure your doc or nurse knows so they can examine you to see if any small lesions ever develop.
If you have any specific worries about HPV, post them here and we'll chat some more.