Of course, work in dry conditions. Buy yourself a hand-held grout tool. I suggest the NON-mechanical one since you can do more accidental damage with it than the hand-held, manual tool. Any Home Center or Plumbing Supply House has them.
Using the tool, remove at least 1/8", or more of the grout. It's not necessary to go all the way down to the 'backerboard' during the removal process. You need to go deep enough to allow the new grout to 'grip' the old grout. I suggest you use a grout that has a hardner and sealer pre-mixed into it. With 'stuff' already pre-mixed, you have enough to worry about. Make sure you follow the instruction completely. Read them first even before you open up the container. This way, while you're at the store, you can get ALL the necessary tools/supplies you need.
Other than that, just take your time and be careful not to accidentally run across the tiles. If you make haste, you're apt to make mistakes. Since you're doing this job yourself, spend some of the $$$ you're saving on good tools and mix. It just makes sense. My best to ya and hope this helps.
PS: There are books at the library that you can read to help with what you want to do.