I've booked 3 or 4 flights on Expedia and have never had a problem. It's sort of convenient because, at least if it's on a major airline that offers seat assignments, you can also see a seat selection chart and pick your own seat. I've never printed out my boarding pass in advance, just checked in at the counter with my photo ID and my confirmation number. But it does seem wise to print out the page with your confirmation number in advance, and keep copies in several places.
It can be sort of a hassle if you need to change your flight later, but that's basically true whether you book online or by phone. Just be sure you read the fine print first, and find out what kind of surcharge there will be if you need to make any changes. And write down as many phone numbers as you can for the airline before you leave, both at home and in your destination country.
Nowadays you should also read all the details about extra fees for checked luggage, arrival taxes, etc. Overseas flights may not charge for the first checked bag, but domestic flights may, they all charge a lot for a second checked bag or any overweight, and the weight limit per checked bag has gone down a lot.
I've had more mixed success booking hotels via Expedia. You can sometimes get very good deals (last time we got a huge $275 suite for something like $90 on an off weekend), but there's very often a catch: lots of street noise, maintenance or cleaning problems in the hotel room, thin walls, annoying nickel-and-dime fees for basic services like internet, that kind of thing. If you have the time and the patience to keep reporting problems to the hotel desk until they meet you halfway, it may work out. But there may also be something to be said for paying a little more and getting more reliable service.
I haven't tried car rentals myself, but from what I read, it's not the online or offline part that matters so much as how busy your destination city is and how early in the day you get there to pick up the car you requested. If you wanted an automatic or a subcompact or whatever and they don't have any left that day, even if you requested one in advance, you won't have much recourse. My guess would be that it might pay off to call or fax the destination car-rental office a few times and get to know the staff before you actually show up, to encourage them to honor your reservation, if you can find a phone number anywhere.
I've also read that overseas car rentals can be a lot more expensive if you wait and do it after you arrive, rather than in advance from home, especially if you're a member of AAA (US), AA (UK), ADAC (Ger), etc. I don't know if that's true, but it might be worth investigating.