I use limes in my cooking, in my morning tea (though for preference I like to use KEY limes there) and during the summer limeade can be very refreshing. Though if your lime tree IS a Bearass, or a Persian, or a Tahitian (the commonest strains) you may find the limeade a little bitter for your taste. I usually hold out until the Creole limes (smaller, shinier, lighter green) come into season. Oh and NEVER try and make limeade with Markut/Kaffir Limes (the wrinkly ones whose leaves you use in Thai cooking). It tastes REPULISIVE. The closest thing I can imagine to describing it is what you would get if you threw a WHOLE lime, (flesh, skin, pits and pith). into a blender, pureed it, then strained the solids out.
You can also use the peel to make your own lime extract. Simply take a grater/zester/or microplane* grate the limes peel into a container of either beverage alcohol or high proof vodka. and let it sit either in your refrigerator or on your counter until the oils leach into the liquid then strain it and bottle it**. The higher the proof of the vodka, the faster and more thorough the job***
*I recommend the third, unlike a grater, a microplane makes it easy to scrape the zest all off WITHOUT getting any of the bitter pith.
** You can simply recycle and empty extract bottles you have, but because I prefer the stronger seal (and I tend to make extract in pretty decent sized batches) I tend to prefer clear or brown glass bottles with screwtops (you can pick them up at the Container Store, on the aisle with the Mason jars and dram/tissue vials.
*** Since you live in White Plains, I'd say the easiest thing to do is what I do when I am making extract, go over to the big wine warehouse near the Westchester County Center (in the shopping complex with the Smashburger and the bowling alley) and pick up a bottle of Devil's Spring Vodka (160 proof). That does the job in a matter of HOURS, and nearly ALL the essential oils come out (the peel bits will actually be WHITE when you strain them out). I'd recommend the same stuff to anyone else making extract, but I am aware that some states do not allow the sale or purchase of drinkable alcohol above a certain proof. For you just get as high as you can.
Oh that reminds me. Because the resultant extract is still somewhere near 160 proof, use it gently. In fact, if you are going the bottle route I mentioned, you may want to get some of the medicine type bottles with the eyedropper caps. And do not use anywhere near an open flame. All liquor is flammable, but about 130 or so, it can actually burn explosively. In fact a lot of online liquor stores have an additional shipping charge for bottles over about 130 proof (not just vodka, a lot of "barrel strength" whiskey can be in the 140-150 range, as can some rums (brandies too probably, but off the top of my head I can't think of any that strong)) because a lot of airlines won't carry anything over that as cargo so it needs to go by ground.