I wrote out some extensive advice on this for one of my friends about a year ago - copied and pasted here. Sorry it's so long, but I think it's good to have as much information as possible and I think that they are fabulous and everyone should be using one, so I get a little excited when people show an interest 
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I just got the basic "right size" Mooncup while I was at uni (their sizing was by age and whether you'd given birth), which served me well for ages and I was very, very pleased with it. But then I didn't have periods for ages because of the pill, and when I started again I had some niggles with my old Mooncup and when I looked online there seemed to be a massive range available that weren't 10 years ago. I'm now using a MeLuna.
So I then discovered that there are all sorts of things to take into account when choosing! But before you get overwhelmed, remember that I started with just whatever was available in Boots at the time and was very pleased with it, so I imagine that the differences that they all make are fairly minor and you can't go horrendously wrong.
A brief breakdown:
*Length - this depends on where your cervix sits. There are instructions on how to determine this on the second website that I've linked below. I'm not entirely sure that I got mine right!
*Capacity - this depends on how heavy your flow is
*Base type - the cups usually have something sticking out of the end which helps to keep them upright and also gives you something to grip when taking them out. The Mooncup has a stem which you cut to the correct length, and this was what I was looking to avoid the second time around. Others have a ball (which I went for) or a hoop, and some don't have anything at all.
*Material type - practically all of them are medical grade silicone or something similar, and unlikely to be made of something that could cause problems, but probably best to check just in case
*Stiffness - some of the cups are harder and some are softer. Harder ones are easier to insert correctly because they "pop out" more readily. (You have to fold them to insert and then the top ring should pop out to expand the whole thing ready to seal and collect blood). Softer ones are more likely to remain folded once you've inserted, which can be a pain because then you have to keep trying and it can be incredibly frustrating. The Mooncup is fairly hard, and I never had a problem with inserting it at all. I didn't even realise that it could be an issue! But the MeLuna, which I deliberately chose to be softer because I was trying to reduce the irritation that the Mooncup had been causing, does cause problems and I find that I'm taking longer to insert it correctly even with loads of practice. My advice would be to go for as hard as you feel you can cope with.
Using one:
I cannot stress enough how much better it is than tampons. When for whatever reason I end up having to use a tampon I can hardly bear how it feels anymore. It just feels healthier, more natural and better for the environment. Plus cheaper! And more convenient because you don't have to carry around extra tampons and guess how many you'll need for a trip etc. However, don't believe all the hype:
*You will probably still need to use pantyliners. I don't remember this happening so much with the Mooncup, so maybe it's because the MeLuna is softer, but I do still get some slight leakage and having panyliners as a backup is definitely worth it. But it's not like tampons where the leaking is an indication that it's full - the leaking just seems to happen regardless, and it's not enough that you have to change the pantyliners very often. It really is very, very slight (more like staining than leaking) and I've never had it "overflow" and need to dash to the loo or anything like that.
(A follow on to that, is that I've also switched to reusable pantyliners. That's a separate discussion!)
*I am not able to go without emptying it all day early on. This depends on your flow, but quite a few people out there will say that you never need to change it at work because it lasts all day. This is unlikely to be true for the first day or so of your period, so you will need to work out how to use it at work. I usually need to empty it once at work on day one and day two.
*Emptying it in public: At home it's simple, because you presumably have a sink near your loo and you can rinse it out easily before putting it back in. In public loos or at work it's harder because there don't tend to be sinks in the stall. Some people take in water bottles, and I happen to have a very small two stall loo on my floor where I can risk darting out to the sink and back again quickly, but if necessary you really can just wipe it clean with loo roll. It's not ideal and a bit gross, but it's fine to do it occasionally and you probably won't need to more than once or twice per period.
*Cleaning it: you need to be conscientious about cleaning it after and before each period. I used to buy Milton's chlorine sterilizer, but more recently I've discovered that you can boil it in the microwave which is much quicker. Don't over boil it, as it could go floppy!
*A logistical issue: The blood, which you empty straight into the toilet, doesn't always flush away the first time and you can end up with some sitting at the bottom which you then need to have a quick scrub at with a loo brush, or try flushing again. What I do is put a bit of loo roll down first, then empty onto the loo roll, then flush straight away while rinsing the cup out in the sink. That way the blood doesn't have time to settle.
First time you use it, don't get discouraged! Give it at least three periods to settle in. If you still don't like it after that, then I'd say give up. But it's awkward and weird and gross at first, and probably difficult to insert easily and will seem like more effort than it's worth. But it's totally worth it! It definitely gets easier with practice. I'd also recommend starting it on a weekend so that you've got lots of time to get it in and get used to it, and to figure out how often you need to change it.
This doesn't feel like a hugely great sell on my part. I'm assuming that you'll get all the positives if you do a bit of googling, and it's better to be honest about the downsides. But even if I haven't gone into them in depth, I really do think that the positives far outweigh the negatives
Resources:
I think that this is one of the blogs I used when I was trying to find a less "stiff" brand, because she has a (very unscientific) list of her impressions of stiffness at the bottom of this entry, and most comparison charts don't seem to include that info:
menstrualcupinfo.wordpress.com/cup-stiffness-comparison-chart/
This website seems to be incredibly extensive, and probably has everything that you need if you want to explore a bit. I've linked to their comparison chart which should be useful for deciding which brand to try:
putacupinit.com/metricchart/
Otherwise, there's tons and tons of stuff out there on Google. The main brand in America is DivaCup, so you'll read a lot about that one, and people comparing other brands to that one.