This depends on a lot of things, particularly where in the US do you live? It may be that you live away from local cedar mills, thus the price of cedar in your area may be prohibitively expensive...
I live in Seattle, and here, 99% of wood fences are cedar. Cedar has a partially effective natural pesticide, which is one of the reasons it is more effective. I would never use pine in a wet environment. With your fence, avoid direct ground contact where possible. Set all your posts in concrete, making certain the concrete completely surrounds the post all the way in the hole. I can't tell you how many rotten posts I have pulled out of concrete blocks that were poured with 6" of post sticking out the bottom. All posts and cross boards should be pressure treated lumber, with only the vertical boards made of your finish lumber. Consider using fence brackets rather than just toenailing your crossmembers to the posts. Your fence will last longer this way.
Make sure you follow your actual surveyed property line. We don't want to have to pull the fence up because your neighbor objects to losing property to you (it really ticks them off) or inadvertantly continue to pay taxes on land you just accidently gave to your neighbor.
A good fence keeps good neighbors. A bad fence creates feuds of mythic proportions.