:welcome: to a fellow Geocacher!
I've been Geocaching since 2005 and a member of CPF since 2007. Combining these two hobbies is great fun!

Below, I will discuss what I have found to work and what I personally use, the reasons behind it, and some options to fit your budget (or come close

).
The two flashlights I find most useful when Geocaching and the ones that always come with me are my
SureFire A2 Aviator and
SureFire E1B Backup. The A2 is hybrid incandescent/LED flashlight in one and the E1B is a two-mode high/low LED flashlight with a nice, tight focused beam.
I would strongly recommend an incandescent as your primary searching light when Geocaching. The color rendering from an incan is very important in the woods (LEDs have a really though time rendering the reds and greens that are ever so present in the woods). In addition, when searching for a Geocache container that is camo'd (either with camo tape or paint), sometimes the color rendering ability of an incandescent can be the distinguishing factor in whether the cache stands out from its surroundings or not.
Next, you really need a low output light as well. This will be used as a backup to your main incandescent, used when signing the log book, and possibly for navigation through the woods when stealth is needed (a little light goes a long ways in the dark woods). An LED is perfectly fine for this low output need.
So with that being said, you can probably see how the A2 is a great Geocaching light with its 50 lumen incandescent beam and low level LEDs built right into the same light. Since these lights are $199 retail and can even be obtained for around $90-110 on CPF Marketplace, I understand that it is WAY out of your price range. So lets look to a cheaper solution, but more in your price range...
What I would suggest is a
SureFire G2 and a
Fenix E01. The SF G2 is an excellent, rough, tough, weather resistant, reliable incandescent with a 65 lumen beam. This will work great in the woods and for searching for the cache. The E01 is a very small, AAA powered LED flashlight which emits around 10 lumens, but runs for around 20 hours on a single cell!
I would suggest putting the E01 on some sort of lanyard or chain around your neck for caching. Unless you like putting flashlights in your mouth, you'll thank me when you are trying to sign the logbook in the dark
This G2 + E01 setup could actually be better than having a single A2 because you would now have redundancy. If you are going to venture into the woods, ALWAYS HAVE MORE THAN ONE FLASHLIGHT! There is saying here on CPF, "Two is one and one is none." Even the best built flashlights in the world could fail on you. Plus its always nice to have the second light to help you find your batteries when reloading in the dark.
I once brought a brand new, high quality flashlight on an in-the-woods Geocaching outing and the lamp died within 15 minutes of use. Luckily I had my E1B with me as well. Keep in mind that the likelihood of this happening is really quite slim, but it can happen (and did happen to me).
The G2 is around $30-40 depending where you buy it and the E01 will run you around $15. I understand that these are still over your budget a bit, but I think these will make excellent Geocaching tools in a great combination.
With whatever you decide on, be sure to check out our list of dealers and their associated CPF discounts to save yourself some money. This list can be found here:
"CPF Specials" webpages/dealers
-Robert