Heh heh- I am sharing all the mistakes I have made in the past to try to give you a fighting chance.
I used to work as a plumbing apprentice and watched plumbers do this install- before doing it myself. The plumbers I watched would lay a thin even bead the length of the seam and then lick their finger before gently smoothing the bead.
As I mentioned previously, avoid putting your finger back in your mouth with silicone on it- tastes yucky and leaves a nasty film on your tongue (I am not aware of anyone who has suffered serious health problems from this and I have ingested my fair share with no ill effects
). Some people use a little water, but it is not quite as slippery as spit (don't know if a little soap in the water would work instead- probably). Dip your finger- smooth a section, wipe well with paper towel and then dip again as your finger dries out.
As a beginner, you might want to do this in 3 sections- eg back wall and then 2 sidewalls as you are likely to be slower than a plumber and the silicone may begin to tack up. Work away from the corners as the silicone bunches up in front of your finger as you smooth it. You don't want a pile-up in the corners. I personally don't like using the masking tape- it leaves raised edges as you pull it off.
As for removing the old caulking- there are products available to dissolve old silicone, but I don't know if it works on acrylic. I have always just used a small putty knife and a lot of patience. If your tub is fiberglass, it would be difficult to use a putty knife without scratching it. Porcelain can take a fair bit of abuse.
Mildew will only grow where it is exposed too water. If the gap is dry behind the tub, mildew won't grow there after it is sealed. Also note, there are grout sealers available on the market. They inhibit mildew growth in the grout seams and make cleaning easier when it inevitably does start to gain a foothold.
Hope this helps!