I think the idea of "a plan" is the best choice here. The idea of getting in and cutting down some trees is most certainly a fun one, but a "willy-nilly" approach isn't what you need. Cut trees need to be moved, and if you can't get access with some kind of motorized transportation, you are going to have a mess. Let's face it, cutting trees is a lot of work. For every minute the saw runs, you are going to spend 5 or 10 minutes dealing with the consequences. That's assuming that you are familiar with how to run a saw and fall a tree. Yeah, logisitics and safety issues, I'm a total "drag" (and not in the cross dressing type), and bringing you "down".
Lay out your access path, and mark out your site. You have the space, do it so it feels right. You would hate to knock down a bunch of nice trees that you didn't need to after the house was built. Yes, trees are fun to cut down, but they are also nice to look at. As a new landowner, you need to look at how to best manage your new little forest over the long term. Cut out the crap, protect the nice trees, and build around that plan.
You may be able to work with the heavy equipment operator to have him knock down a few trees, then move to another part of the plot while you cut off the root balls, then have him push the root balls to one pile, and the logs into another. We had a construction guy at one time who would pull out the logs he thought we would like and have them laid out in a real nice order for us. We liked him. It can be done. Just don't be in a hurry. This is a real opportunity to have something nice. Think the process through, and think about how you are going the spent that extra 10 minutes of work for that minute you just ran the saw.
Now, as someone else mentioned, try renting a saw, or going to a GTG to get some "hands on" experience. You are going to need one, and realistically, one in the 50-60cc range. Look at used, look at Stihl, Husky, or Dolmar/Makita. I've never owned a new saw in my life, and probably never will. Too cheap. The GTG is a great idea, since you will get not only a chance to run some saws, but a chance to learn how to do stuff...sort of. Then there is the beer part, but that comes later. Hell, if you wish to get really cheap, find a good Homelite XL12 or Super XL for $50 and cut away. I paid $20 for mine, and have run the crap out of it. No chain brake, loud, slow, and heavy, but it has to be one of the most satisfying saws to run ever. It's just so "right". Perfect saw for "lot maintenance" and for the times you just have to cut something. Some will argue this point, none will have ever run one.