There is some very good advise in this thread. I second all of it.
I have had bits come out of the router before ... cheap junk bits. In one case, i had one actually fail and come apart (one cutter flute came off while routing). The resulting imbalance tore the bit out of the chuck, broke it into three or four pieces, and sent them flying across the shop. One piece hit my face shield (I always use a full-face shield when routing). The others, thankfully, missed me. One was stuck in the wall. I never went near those cheap bits again. Those Chinese bits from the discount tool vendors may seem like a good deal, but they aren't worth it.
Get a good router. Get good quality bits. Learn to comfortably and confidently use the tool. Practice first. Wear all the safety equipment and never shortcut safety procedures. Never rush the job. If it doesn't feel safe to you, it probably isn't -- so step back and re-evaluate things first.
When working with ANY power tool, always think first about what would happen if the tool were to fail. Where would the cutter go if it broke -- make sure none of your body parts are there. For example, never put any body parts directly in line with the blade of a table saw (as much as possible anyway). If the blade fails, it will (usually) leave in the plane of the blade.
And after you've done all that ... enjoy your router because it's a wonderful tool.