It can be a bit daunting using a cup the first few times. If you aren't familiar or comfortable with your body, it might be helpful to get yourself acquainted first. Don't be shy! The following are some tips and advice collected from other cup users.
Insertion and Removal
Make sure the rim fully pops out when you've inserted the cup, to ensure you have a proper seal. (You can feel this if you run a finger around the cup once inside).
Squatting is a recommended position for insertion/removal. I personally find a reclining position on my side to be easiest. Some women find standing with one leg on the bathtub/toilet seat works for them - try several positions to see what is easier for you.
Remove the cup slowly, and make sure you have broken the seal, don't just yank it out.
Let the cup open as soon as the rim is in, then push it up further if you need to, or do a few kegels to move it further up.
Turning the cup (Hold the base of the cup and turn it) can help it fully open and create a seal.
If you are having difficulty, try a different fold.
If you can feel it, or its uncomfortable its probably in the wrong spot.
It doesn't go in as far as a tampon, it should sit just inside the vagina.
If it gets "lost" don't panic. Relax and do some kegels (pelvic floor exercises - almost like trying to do a bowel movement) to push it back out.
Because they don't absorb like tampons do, you can have a "dry run" to test it out before your period so that you are comfortable using it before you need to. Although until you are bleeding you won't know if it is working properly, it is helpful for learning insertion and removal and getting to know your body.
Other Tips for Use
To clean the airholes, fill the cup with water, place your palm on top, turn this upside down so your palm is over the cup and squeeze the cup - this should force water through the holes.
Don't let it get overfull, as emptying could then be messier.
Carry a moistened cloth or a bottle of water with you if you need to empty the cup while out.
Trim your fingernails before using the cup.
You may need to trim the stem (But try with it uncut, and don't cut it all off in case you have to return it - as some manufacturers won't take it back if the stem is cut off)
Emptying it in the shower (while showering) means you don't need to worry about spills.
When emptying the cup into the toilet, flush straight away, put toilet paper in the bowl first, or swab some toilet paper around inside the cup to soak up the blood then toss the toilet paper in the toilet - otherwise the blood will sink to the bottom and hard to flush out.
Don't be alarmed if the contents of the cup isn't all liquid blood - there may be clots and clear liquid in the cup too.
There is a large live journal community devoted to menstrual cups, where you can post questions and seek others advice - however... there is also a large memories section where you can look at previous posts because chances are the questions you have will have been asked many times before. Here is a link to that memories section directly.
I use the "C" fold. I fold the cup in half (with both hands), and hold it in that position with my first finger and thumb - running down the length of the cup, so that my fingertips are at the rim end.
Here are some videos about Cups you might find useful