1. INSERTION - The more complicated the fold, the springier the cup will be, so the easier it will be to pop open. My favorite is the half diamond (http://www.youtube.com/watch?v=6Cdc3VaVgJM). It's the smallest fold I've found, and it pops open consistently. As many other posters have noted, insert the cup with the diamond facing down and push straight back.
2. STEM - If the stem is bothering you, do some kegels and push the cup up as far as it'll comfortably go. Is the stem still bothering you? Can you remove the cup by grabbing the base, not the stem? Then remove the cup, wash it, and trim down the stem with clean scissors or nail clippers. Repeat. I ended up cutting the stem off my Lunette Selene size 1 completely. There was no way I could've made it through a whole day, let alone a whole cycle, with that thing.
3. C-FOLD REMOVAL - The first couple of times removing my cup hurt like hell. Then I tried folding the cup as I removed it, and it was much better. Grab the cup by the base and pinch to release the suction. Then pull out slowly, pinching the cup into a C-fold. (Make the C point up, unlike when you insert it.) That way the cup is smaller for removal. I usually manage to do this without spilling by bringing the base down so the cup is upright and letting it come open upon removal.
4. CLEANING - When you're at home, remove the cup and wash it with warm water and baby shampoo or non-antibiotic hand soap. To clean out the holes, fill the cup with warm water, cover the top with your hand, and flip it over. Squeeze the cup to push water out through the holes. It's OK to remove the cup, wipe it carefully with TP, and reinsert it if you're in a public restroom. Just be sure to clean it thoroughly when you get home. At the end of your cycle, boil the cup for 10 minutes. Leave it out in the sun for a day or two if it smells funny. Then put it back in its pouch or in a non-airtight container until your next cycle.
Most of these tips are summaries of what others have shared on this site, but I believe the C-fold removal is new. Enjoy!