You didn't say where you are from, and that matters a lot. If you are in a big city your options become more clear. Out in the country your options are more limited.
If you are in or close to a decent sized city, just go through Craig's List and find comparable for the vehicle you are selling by searching the listings. That is your competition. Next go to kbb.org and edmunds.com and find the value for your vehicle. Based on all of that data, figure out what you would be happy getting for your vehicle. For simplicity I would consider a number half way between private party value and trade-in value as what you should expect. But I would list it for closer to private party value, especially if you are looking good against the competition on Craig's List.
Obviously it would be great to sell first, you really don't know how quickly you will sell or for what amount, so if you can borrow a vehicle for a short while I'd go with that route. We just traded down from a truck to an Outback, and we purchased first because we couldn't borrow. But if you do that make sure you aren't counting on some unrealistic number to get when you sell your vehicle. Just like selling a house, pricing it right is important. If you start too high you are just setting yourself up for disappointment and delay, and you probably won't get more than if you start at a fair value.
I agree with the private party idea, you will pay a premium working with a dealer. Not that it's always the wrong move, and it's definitely more simple, but if they weren't making good profit they wouldn't be in business. They will happily give you cash back on a trade in, but not as much as selling and buying yourself.
Regarding purchasing from a private party, the key is to set rules and follow them. Rule number one is to have an independent mechanic look at the car before purchase. Driving 1 mile around the sellers house is not going to tell you if it's a good vehicle. You will have to pay for the inspection, but it will save you a lot in peace of mind and potential big ticket costs. Just let the seller know ahead of time and line up a local shop to do the inspection. It doesn't have to be super thorough, just a good overview. If there are possible issues you can either walk away, or use them as negotiation points.
Number two is to have a ceiling before showing up. Don't get so set on a vehicle that you keep raising your offer since you just want to have it over with. Based on your research on kbb.org and edmunds.com you should have a really good idea of value before you show up. don't pay more than private party unless there is a really good reason. I offer trade-in value, and work my way up from there. I consider half way between trade-in and private party value to be a good deal, but it can vary a lot depending on how many of that vehicle is available right now.
Just my two cents, hope it helps.