Silicone is the correct choice, but use GE or equivilent, not the cheap stuff. The "cost" of this project, or any project really, is the pain in the neck of having to do it.
When you purchase your caulk, ask the salesman for a razorblade window scraper, which is a small plastic tool that holds razor blades. Use this to scrape the old caulk out, being careful to keep the angle of the blade lying almost flat against the surface you are scraping. You should be able to pull the old caulk out with one hand while you scrape with the other.
Once you get most of the caulk out, you must thoroughly clean the area behind and around the original caulk. Purchase a 12oz can of Goof-off and use an old rag or washcloth, with rubber gloves. be certain to turn on the bathroom fan and DON'T SMOKE AROUND THIS STUFF!
Have your store salesman show you how to apply the caulk and smoothe it. Tricks: Purchase a caulking gun that relieves the pressure on the tube when you stop squeezing the handle. These are called "no-drip" and run around $12-$15 or so. You will appreciate the extra cost is worth it. In addition, you will smooth the caulk with your finger, but keep your finger wet when you do it. Otherwise, the caulk will stick and won't smoothe as nicely.
Keep in mind that you cannot paint over silicone caulk. Do not use this product in an area you intend to paint at a later date.
One last item. Consider sealing your grout on your tile shower, if it is tile. Allow the shower to dry by not using it for at least 2 weeks. Then clean the grout and apply a silicone grout sealer. This repels water, reduces mold and mildew stains, and increases the useful life of the tile.