Make sure you have thoroughly removed the old caulk around your bathtub before replacing it with new caulk. You can buy a scraper tool at any hardware store that will help you easily scrape away and remove the old caulk.
When you are purchasing new caulk for your bathtub, you will want to purchase an anti-microbial caulk because this will prevent mold from forming or coming back in the future.
Before applying the caulk, you will need to cut the tip off the applicator. Don't cut the tip too big because you don't want too much caulk. Try to cut the smallest hole possible on the applicator. You can always widen it later if necessary.
Make sure the joint where you are applying the caulk is clean and free of debris. Then, start at one corner, and squeeze your bead of caulk out, evenly running it from one corner to the other. After you have evenly squeezed the caulk into the joint, dampen your finger and smooth down any beads or bumps. If you accidentally put too much caulk into the joint, just take a damp rag and run it across the joint, wiping up any excess caulk.
You should now have a smoothly caulked joint on your bathtub.