AP can definitely be done with just an EOS and lens, especially when starting out.
The 60D is a good camera but do bear in mind of the weight of the camera. Initially this won't be an issue if you're just using cam+lens on a static tripod. But when you start mounting it on a telescope, the weight of the 60D starts to be a consideration, especially if the telescope is light. Which is also why many DSLR APers still go with the Rebel (XXXD or even 1000D/1100D) route. That doesn't mean the 60D can't do it cos it definitely can.
For the Milky Way, you'll want a wide lens. For starters, your DSLR may come with a basic kit-lens of either 18-55, 18-135 or 18-200 or any variant. That's a good place to start. Here are some of my starting settings when I first tried it out:
1. Set camera to Manual (M mode)
2. Shutter depends on focal length (I use the formula 400/focal length to determine)
3. Set aperture to around 800 or 1600 for starters
4. Use the remote timer (2 secs) or a remote shutter release if you have
4a. Alternatively, hook up your dslr to your laptop/pc and use EOS Utility to fire the camera
5. Turn the focus to infinity or near infinity to roughly get focus.
5a. Use the Live View function and Zoom in (5x or 10x) to finetune the focus
6. Fire away!
For the above, polar alignment isn't important, since the tripod is static and the shutter speed is calculated to minimise any star-trails from appearing.
I use Stellarium, a free program to get an idea of the stars that are on display. You can also search and read up on stacking software available. Most AP photos are a result of multiple shots stacked and combined together to capture and enhance the light photons captured per shot. I'm currently using DeepSkyStacker - which, in itself, is another journey in using and learning.
Once you move to longer exposure AP, then polar alignment becomes critical since you want to move beyond the shutter limitations in Step 2 above. Minimising vibrations and movement errors also become another consideration.
I definitely agree on the stability of the mount as most important when moving to the next stage. Some will say that a CG-5 GEM is the minimum but it also depends on the size and weight of the telescope. E.g. I'm using a CG-4 GEM but because my telescope is an Astrotech AT66ED which weighs only 1.8kg, it's sturdy enough for me. Mounts are expensive in SG (or at least, to me) but even then, when I shipped my CG4 from overseas, the shipping almost killed me! So you have been warned.
AP is a long dark road that's daunting and challenging. I'm still a beginner in every sense and learning as I go along but once you get a taste of what you can do...... then that's it. Hook, line and sinker!
Good luck and have fun in your journey!
EDIT: You don't necessarily need any filters to start on AP. Also note that if you buy a clip-on filter and put it into your DSLR, you cannot attached EF-S lenses to it. This is important unless you are only using EF lenses. (Just in case you're not aware, the 60D is an EF-S camera). Filters are there to reduce light pollution, enhance colours etc but does not mean you can't capture nice pics without it. Try it out first, especially if you already have a camera system. If you don't like it, it wouldn't have cost you anything!
