Both the laser collimator and cheshire eyepiece are tools used in astronomical telescope collimation. What are the main advantages/disadvantages of each? Which one might be best suited to start with for telescope newbies?
Why not both? (if you can afford them)
Both will do the job, but each has its strengths.
Cheshire eyepiece (or, more properly, Cheshire/sight tube combo, which is the popular combination nowadays):
It's easier to use when the scope is grossly out of alignment. Like, when you put it together for the first time, or after it dropped on the ground and rolled downhill for a minute. But can you do that with a laser? Yes.
It's easier to align the secondary mirror, and it tends to produce dead-on results for this purpose. But can you do that with a laser? Yes.
It is not immune from precision issues. Most parts of the Cheshire are usually well aligned (because they are machined on a lathe), but the crosshairs on the sight tube may or may not be perfectly centered. But can lasers also suffer from alignment problems? Yes.
Recommended Cheshire / sight tubes: the ones made by Catseye.
They make much more than just Cheshire eyepieces, and all their tools are very high quality.
It's faster and arguably easier to use for normal daily collimation before you begin your normal observing sessions. But could you use a Cheshire instead? Yes.
A laser is only as good as its own alignment precision. A misaligned laser will not collimate your scope properly. But do Cheshire sight tube combos suffer from precision issues with their crosshairs? Yes.
Recommended laser collimators: the laser collimator and the tuBlug combo made by Howie Glatter. Unfortunately Howie has passed away recently, so either wait until someone picks up where he left off, or find a used laser/tuBlug combo somewhere on eBay or classified ads.
Howie's lasers are essentially guaranteed to have zero collimation errors from factory. From anecdotes told by users, these devices tend to keep their perfect collimation even after being dropped to the ground repeatedly.
The tuBlug is not mandatory, but it makes primary mirror alignment much easier.
Keep in mind that you also need a precision center mark on the primary, for these methods to work well. Hopefully it's been placed there precisely from factory; otherwise there are ways to re-center it.
How to verify if the laser is centered:
Place it in the focuser. Do not tighten the screws too much, allow the laser to rotate in the focuser a little bit.
Turn the laser on. Watch where the spot hits the primary. Now rotate the laser in the focuser. If the spot describes a circle as you rotate the laser, then the laser is misaligned.
Different method: Make a V-shaped trough from 2 pieces of wood or aluminum. Placed the through horizontally on a sturdy table. Put the laser on the trough. Turn it on, projecting the spot on a distant flat object, like a wall or fence across the street.
Now rotate the laser in the trough and watch the spot. Does it stay put? Then the laser is fine. Does it move in a circle? Then the laser is misaligned - find the adjustments screws (if it has them) and adjust it until it's perfectly square.