Gun season is obviously the high reward part of the year. Depending upon where you hunt (here in Ohio we can only use shotguns and muzzleloaders), your shotgun could be converted into a slug gun for as little as 5 bucks and an IC choke tube. Although rifled slugs don't have the ballistic capabilities of a rifled barrel shooting sabots, they're definately an economical way of getting you out deer hunting.
That said, there's far more accurate methods to pursuing deer, such as a rifle (obviously) or a cantaliever rifled barrel for your shotgun with Hornady SST slugs. If you want something with a better performance than a shotgun, but are willing to sacrifice some on the range, go with a muzzleloader. You'll be able to shoot out to around 200 yards with fairly flat trajectory, and even push the limit to 250 or so with increased drop. This also allows you at least 1, if not 2 extra weeks/weekends for hunting with a gun, as there are separate muzzleloader seasons in every state (as far as I'm aware).
For bow hunting, you can get a fully decked out bow for as little as $300 new. It's going to be slow, noisy and ugly, but you're still going to have a blast with it. If you can't commit a lot of time to hunting, I suggest not even wasting money on a bow. If you're ready and able to spend countless mornings and afternoons in the treestand, then I would highly suggest investing in a bow. The excitement and anticipation of hunting the rut, coupled with the satisfaction of taking one of these beautiful animals with a couple of sticks and a string (yes, I realize we've progressed far beyond that) makes bowhunting whitetails an unrivaled hunting experience. You simply cannot get the same experience from any other type of hunting, and if you are fortunate enough to take one come September or October, you'll understand what I'm talking about.
My suggestion is to get on the classifieds here, and look at gun broker. Buy yourself a beginner bow as well as a gun, and take advantage of both awesome opportunities. If you fall in love with it (which I have no doubt that you will), then you can upgrade your bow. Until then, no sense in dropping a grand on something you may never use again.
Here's a few links to some package bows. They're a great way to get started. They're not the fastest or quietest, not even close. But I'd rather bow hunt with one of these than not bow hunt at all!
http://www.basspro.com/Bear-Archery-...04466/-1658791
http://www.basspro.com/Parker-Bucksh...9#BVQAWidgetID