Any tips for caulking around a tub?... what's the best product to use?

For mold, spray with vinegar.

https://www.thisoldhouse.com/how-to/how-to-caulk-around-tub

Don't know what kind of caulk you're using, but use silicon with a mildewcide in it. https://www.youtube.com/watch?v=eLnVB-u0txs&t=246s they also make a silicon caulk solvent that helps soften it up.... https://www.homedepot.com/p/Motsenbockers-4-5-oz-Silicone-Latex-Caulk-and-Foam-Sealant-Remover-411-45/100395888

Preparation is key!! I cannot stress this enough. You must remove all traces of the old caulk, and by all traces I mean ALL! So after you've gotten all the caulk off the tub and the tile(or wall if its painted) go back with 150 grit sandpaper and rub the clean areas (tile only, no sandpaper on paint) to see the film you didn't get. That film will prevent the new material from forming a bond and creating a seal. If the wall is painted, use paint thinner on a rag, test for color fastness, and if its ok, wipe down the wall. Remember! this is for a bead of caulk, we're talking one half inch of space on the tub edge and on the wall. Now for my personal secret. Lay down a strip of blue painters tape on the tub, about 3/8 of an inch from the seam where the tub and wall meet. Its not rocket science, you can approximate. Repeat on the wall.

Now lets talk about caulk. Caulk talk, get it? There are many, many types and brands. This is not the time to go cheap. Look for a water based silicone, and look for one that never "hardens". One tube should do, so by two. If you don't, you'll need it. Its a law of the universe. Cut the tip on an angle, look for cutting guide lines in the plastic of the nozzle. Make a continuous bead from front to back, then from inside corners to outer edge, and if you've peeped it, go down to the floor. Next, with cloth rags at hand, use a wet finger to smooth the bead down to form an inside curve that will allow water to run down the wall and into the tub. There will be excess, wipe it on the rag, rewet your finger and continue.

If you've done all the prep and gotten a nice bead, and smoothed it out with your wet finger, theres nothing left to do but remove the tape. Do it slowly, as this will leave you with a perfectly finished edge that tapers down to nothing. Good luck.

A word about mold. Its always there, waiting to grow. best way to prevent that is with peroxide. It kills everything microbial instantly, you can watch the black disappear. Open the cap, but don't peel the inner foil seal. instead pierce it with a pin 3 times close together in the center. You now have a controlled release, squeeze the bottle and aim the 3 streams at the seam about once a month, you'll never see mold again. Also recommended for germ killing in the kitchen, and anywhere you want to disinfect where there are children, as it is perfectly safe to touch or accidentally get in the mouth. Peroxide is H202, water is H2O. It turns to water as it evaporates. any residue is water!

Stop in Homedepot, and ask for the right caulk. You have to make sure you get all the existing caulk out, let the area dry then caulk, and let it dry completely.

http://homeguides.sfgate.com/type-bathtub-caulk-91522.html

You have to use the proper tub caulk so that it does not crack or mold.