One issue that's going to have a major impact on your wide-field imaging is the amount of light pollution you have at your site. It's not only the brighter skies but the gradients you will get when trying to image over a large area. So, if you have dark skies with little to no light domes from surrounding cities then it's possible to do some really amazing work with just a wide-field, normal, or short telephoto lens. With light pollution you probably want to keep your targets pretty high in the sky, maybe within 30 degrees of the zenith (for really good results).
As for exposure times, 10 to 30 seconds is a little optimistic unless you are using a very wide angle lens. But, it depends upon how sharp and round you want your stars to appear (and how closely you look and how large you want to display or print the results). I use a "rule" of 100 divided by the focal length of the lens and that gives you a good starting point for a maximum exposure in seconds. I think the traditional (and often repeated) 500 "rule" is outdated and really more appropriate for film-based cameras, not digital imaging with today's small-pixels (but, YMMV).
You can expose longer when imaging closer to one of the celestial poles, but if you want really pin-point stars and if you are imaging near to the zenith then the "rule" should probably be something like 50 divided by your focal length (that means a maximum of 1 second for a 50mm lens). That said, with a wide-angle lens and when imaging the Milky Way you can still try exposures in the 10 to 30 second range, but don't expect them to look sharp in anything other than a 4x6 inch print or on you phone's small display.
I'd say start with what you have before spending any more money. Also, don't overlook the possibilities of buying a used, manual focus lens from a place like ebay. Some of these older lenses can be purchased in good condition for around $100 (U.S.) and they can produce good results as long as you close them down by a few f-stops.
As for using a tracking mount, yes, that can help greatly but in my experience the performance/quality of the low-end camera trackers varies a LOT from sample to sample, so you never really know what you are going to get (some samples are pretty good, some are very bad and it probably makes little difference in the make or brand).
In any case, as I suggested earlier I'd just start with the equipment that you currently have, which means just about any camera body, a lens that has a manual focus ability, and a tripod. You may also need some type of remote camera release or intervalometer, since you don't want to be touching the camera when you start your exposure. I don't know whether your Rebel T3 has a built-in intervalometer, but many cameras can be used with an accessory intervalometer, or with software like BackyardEOS or BackyardNikon, etc.
For image samples and other tips on technique, you can do a search here on CN. Here are just a few samples from my own work:
(tracking mount with camera lenses)
And below is an image of the constellation Lyra that I took with a Nikon D5100 DSLR and a Nikkor 50mm f/1.8 Ai-S pancake lens set to f/4 (ISO 400, a stack of 32 images, each exposed for one second, producing a total integrated exposure time of 32 seconds). No guiding or tracking was use, just the still camera on a fixed photo tripod. You can even see the Ring Nebula in this image, if you know where to look, it appears as a very faint star. This was done under red/orange zone light pollution, but when Lyra was near to directly overhead.