So, the pergola we want to build was quoted around $5000 for the 12'x16' area. Any idea on how much the lumber would cost for this kind of project? I have little experience, so I am wondering if this is a little too ambitious as a DIY project.
If you get a decent set of plans, it's not too ambitious.
However: it sounds like you probably haven't done much woodworking, since you're not sure how to price lumber. That implies that you probably don't have much in the way of tools. It also implies that if the plans aren't quite right, you won't have much experience in figuring out how to fix them.
That's not to say you shouldn't do it. Everybody gets started somewhere! But it means that, in addition to the lumber, you'll likely need to buy tools and things that some people here already have. If I were going to build something like that, I'd probably use the same timber knotscott did, which isn't exactly cheap. Cutting the ends of the posts straight can't be done trivially with a circular saw (it's doable, and not really that hard, but it takes some practice), so a chop saw would probably be a good idea. You'll also probably want to set the posts in cement foundations of some sort.
None of this is hard, necessarily, but it takes a certain amount of thought to figure out how to do it. You may find that the kit more than equals the investment of time and money you have to make to do it on your own, or you may not... that depends a lot on how much time and money you have at the moment.