Welcome to The Helpful Gardener and Happy Birthday!! Sounds like a lovely present. You don't say what type of rose you have, but since it's potted and you're keeping it indoors, I suspect it's a mini rose. This site has all kinds of info on how to care for it.
The small pot has peat moss wrapped around the base of the bush. I'm guessing I need to leave that in place?
Maybe you mean sheet moss? It looks like the moss you would find in the woods. I would suggest you remove that so you can better see the soil and it will be better able to breathe.
Somebody advised me to place coffee grounds on top of the soil. I have not done this yet. Should I do so?
It certainly wouldn't hurt and would be good for your rose to put USED coffee grounds on top of the soil. Just a bit so you still have room to water.
How often, how much do I water?
Water when the top inch of soil is dry. Just poke your finger into the soil up to the first knuckle. If the soil is dry it's time to water. The more sun the plant gets, the more roots in the pot and a clay pot will all mean the soil will dry more quickly. Water until the water runs through the drainage holes and don't let it sit in water for more then 15 minutes or so.
Is there a need for fertilizer and if so what kind is best?
Roses love fertilizer. If you use the coffee grounds you could fertilize once a month in the fall and winter and twice a month in the spring. I prefer organic fertilizers. Synthetic fertilizers like Miricle Gro leave behind residual salts that can accumulate in the soil. Fish emulsion or fish emulsion mixed with sea weed is great. It doesn't smell good but the smell dissipates after you water. You could even use milk. Yup, the stuff you put in your coffee. I've been doing that with my houseplants for years and they love it. I put about a tablespoon in a quart watering can. Since you only have this one plant, when you finish a container of milk add some water and use that to water your rose.
You should also find this site about potting houseplants helpful.