There's really two separate issues here... one is about focus, the other is about shutter speed. Both can cause blurred results, but they have different answers.
For focus:
Running the focus all the way out on the lens isn't trustworthy... cameras can focus "beyond" infinity because, for example, if your lens was being used to shoot infrared (and you can do an IR conversion to a DSLR camera -- or shoot IR film on a film camera) then the focus point is actually "farther" than it is for much of the visible light spectrum. Lenses are designed to allow that (people do shoot IR). You also usually can't trust the distances marked on the lens to be bang-on accurate (though they usually are close).
I find the only way to focus is to do the following:
1) Disable auto-focus by switching the lens to manual focus mode.
2) Enable "live view"
3) (optional) set the ISO to max (e.g. ISO 6400 for your 60D) and set the shutter speed to 30 seconds (this is because your camera has a feature called "exposure simulation" which is enabled when using live view by default. This will cause everything to appear brighter so that many stars should now appear bright enough for you to tell if you've got them in focus.)
4) Find any bright star (even if it's not in the part of the sky that you want to shoot - if anything out in space is focused, then "everything" out in space is focused.)
5) Zoom in to the 10x magnification in live-view
6) Very carefully adjust the focus ring to get your stars down to pinpoint size.
7) If you changed exposure in step 3, then don't forget to return the camera exposure to more reasonable settings. I only crank the exposure up in step 3 to make it easier to focus.
You can now switch off live-view, re-compose the camera on any part of the sky and begin shooting.
There is an alternate way to focus. You can use something called a focusing mask... the most popular being the "Bahtinov" type focusing mask. It covers the front of the lens and has lots of slots cut into it on various angles. These slots cause all stars to display "diffraction spikes". The mask is designed so that you'll see two spikes that cross each other like an "X" with a 3rd vertical spike "|" that crosses somewhere through the X. If all three spikes converge at a common center point, you have nailed focus.
The spikes are easier to see in longer focal length lenses or on very bright stars. Dim stars don't throw very big spikes -- so they don't make good focusing targets when using the mask.
You can either buy or build your own focusing mask. The Astrojargon.net website has a mask generator (you punch in lots of parameters about your lens and it creates a template. You then download the template, print it out, trace it onto a durable material, and cut out the slots. Commercial masks are made by vendors such as Spike-A, Kendrick Astro, FarPoint Astro, Gerd Neuman, and I've somewhat recently picked up the "SharpStar2" by LonelySpeck (I haven't used it yet). The SharpStar looks to be the most interesting for normal camera lenses because it allows in more light (making stars brighter and thus easier to focus) and it's a square drop-in type filter (you slide it into a filter holder such as a Lee filter holder or a Cokin filter holder and you can put these filters on most any camera lens (as long as it has filter threads on the front of the lens.)
You do not _need_ the focusing mask... I'm just offering it as an alternative way to nail the focus.
For exposure:
The stars appear to move across the sky from East to West. The effect is called "diurnal motion" and it's caused by the rotation of the Earth. Your human eye doesn't notice it unless you wait a long time, but a camera lens on a fixed tripod sure WILL notice it and the result will be elongated stars (instead of pinpoint stars ... your stars will stretch and grow into star trails.)
If the exposure duration is short enough then the stars will not have moved enough to start showing star-trails and you'll still get pinpoint stars. But the question is... how long is that duration?
The answer depends on the camera sensor size and the focal length of the lens. There's a long version of the formula where you do some math to determine how many arc-seconds of sky fit into a single "pixel" on your camera sensor and then based on how large you plan to display the image (and from what distance will the observer be looking at your image) then how much movement would it take before a person would notice this. (The Earth rotates at a speed of 15.04 arc-seconds per second of real time).
But there is an easier way... you can just follow the "600 Rule" or... be a bit more conservative and use the "500 Rule" instead.
The 600 Rule suggests that if you have a 35mm film camera (or a full-frame DSLR) then you can simply divide 600 by the focal length of your lens. The result is the number of seconds you can expose and not get star-trails.
In other words, if you had a 20mm lens then 600 ÷ 20 = 30. That means you could shoot a 30 second exposure and get good results (with a 20mm lens on a full-frame camera). If, however, you had a 40mm lens then you could only expose for 15 seconds. If you had a 200mm lens you could only expose for 3 seconds.
Some people feel that using 600 as the base value is a bit too generous and that upon close inspection they can see some elongation in the stars. So they use 500 as the number to be a bit more conservative. If you're producing images to be displayed on the web (this pictures won't be very large when displayed) then 600 is fine. If you plan to make a very large print and hang it on your wall... you probably want to use 500.
But you don't have a "full frame" digital camera. Your 60D has an APS-C size sensor with a 1.6x crop-factor. You can either multiply the focal length of your lens by the crop factor... or divide the base by the crop factor (either way gets you to the same number). E.g. with 20mm lens, you'd multiply that by 1.6 to get to 32. Divide 600 ÷ 32 = 18.75 seconds. Your camera doesn't have a shutter speed for 18.75. Probably you'd round downward and take a 15 second exposure. You could try to round up to 20 seconds and see if you like it (but you might see a tiny amount of elongation in the stars.)
What about "long" lenses?
If you want to use long focal lengths (such as 200mm) things get a bit more complicated because the shutter speed has to be shorter and now it's difficult to collect enough light. So you end up cranking up the ISO, but that results in very "noisy" images.
You have a few choices... you can:
a) Use a tracking head for your (hopefully very sturdy) tripod. E.g. use a Sky Watcher "Star Adventurer" tracking head. I use a head called a Losmandy StarLapse. There's also the iOptron SkyTracker head. The tracking heads are aligned so that their rotation axis is parallel to the Earth's rotation axis (so they're pointed toward the celestial north pole... not the camera... just the tracking head. You can put a ballhead on the tracking mount and point the camera anywhere you want.)
b) Take LOTS and LOTS of short exposures and use stacking software designed specifically to align by stars. Deep Sky Stacker is free and I'd start with that. I use a commercial product called PixInsight which is not free, but it's very powerful (warning: PixInsight has a long learning curve and you'll find it much easier to learn with something like DeepSkyStacker.)
c) Use a shorter focal length. Shorter focal lengths are always easier because they are more forgiving when it comes to tracking precision. I usually suggest people start with short focal lengths (wide-angle lenses) simply because these focal lengths are more forgiving on both exposure duration and tracking precision. That way you're not trying to learn how to overcome too many obstacles all at once.
