If the flights were all booked via the same airline, or as part of the same itinerary, there's no problem. Normally they'll hold a plane a bit longer if they expect connecting passengers, and if you're REALLY delayed, you'll be checked onto the next flight. Your luggage will be offloaded with you as well.

I strongly suggest travel insurance anyway, if you haven't already got it. If you get severely delayed and/or your luggage goes walkabout for a while, you will have your accomodation paid for (although the airline will pay, normally) and you can claim for reimbursement of extra expenses (such as buying clean underwear/toiletries while you're waiting to be reunited with your luggage). You can get some very reasonable deals online - a few dollars a day for peace of mind is worth it, I think.

If you've booked your flight with credit cards, they often give you some form of travel insurance. I'd check that out too.