To remove caulk, try cutting flush with a razor blade. Scrape away as much of the old caulk as possible without scratching the shower surface. You don't mention what the shower surface is made of??? Then clean the surface with a rag and cleaner like rubbing alcohol removing any additional small peices of caulk. You could also clean with acetone depending on the material your shower is made of?
The key to this job is taking your time to remove all old caulk and preping the surface properly. The surface must be clean, dry, and free of soaps, oils, or anything else.
Once it is throughly clean & dry, Apply the new caulk. I would recommend the 30 year or 50 year stuff with pure silicone. Apply a smooth even bead with a slow but steady motion. This is the hardest part for me. Then let dry for 24 to 48 hours.
Finally, mold and mildew will reappear regardless of what you do. I keep a small fan in the bathroom which I set blowing in the shower stall after I am through. In 15 minutes, the shower is comlpetely dry. So far, I have prevented green & black stuff from growing for 2 years.
Good luck, and remember to run a bead that is water tight!