Halter classes can be a great way to get started in showing horses. Depending on where you're located, 4-H may offer/require 2 different in-hand classes: Halter and Grooming and Showmanship. Both look similar from a spectator standpoint, but they are judged differently. In Halter, the focus of the judging is on the horse. Lolly will be judged against the ideal 3 y/o Paint mare in terms of conformation, typey-ness, and movement at the walk and trot. If you find yourself in a Grooming and Showmanship situation, the bulk of the judging emphasis is on you, and Lolly is essentially a prop for you to demonstrate your hard work in grooming, conditioning, training, handling, and showing her to her best advantage. G&S usually has a trickier pattern, meant to test your ability, as opposed to relatively simple Halter patterns which are meant to demonstrate your horse's movement and carriage in gait.
Common entry-level maneuvers for both classes are walk, trot, halt, back up, turn on forehand/pivot (always away from you), and set-up. There are a couple more things that a judge may ask for, but not typically until intermediate/advanced level G&S. The judge is looking for a fluid pattern and a willing, in-tune horse. If you know that Lolly pulls you forward, or that you end up dragging her behind, that's something to prepare.
For a laggy horse, I like to carry a long whip (either dressage or lunge, depending on how long of a horse I'm working with) to back up my cues. Ask, suggest, encourage. You take the first step forward (or up in gait). Make it exaggerated a bit, so Lolly can't miss your change in energy. If she doesn't match your step off your rise in energy, use a vocal cue, such as a cluck or "walk," etc. If she still doesn't get the message, tap her hip with the whip. Don't look or contort yourself around to do it, just flick your wrist so she gets a little pop. That's plenty of pressure for most laggy horses. For a dragger, I do lots of transitions between walk and halt (eventually trot and walk, and as a real test, between trot and halt), backing up after each halt. The ideal is for her throatlatch to be at your shoulder without you having to constantly bring her back or pop her up.
Pivots and set-ups/square-ups are a bit trickier, and take practice. There are some great posts and threads in the Horse Showmanship section on teaching and polishing those movements. You'll need to show her with a chain, whether she "needs" it or not. She shouldn't need it for control, but it does allow a very subtle cue when it comes to backing and setting up. The chain lead should match your halter. Practice with the chain at home, so it isn't a surprise to her on show day.
In terms of grooming, you need to start with a clean horse. As Lolly is a Paint, I'm guessing you'll have at least some white to deal with. If the weather is warm enough, a bath is in order. There are some awesome whitening shampoos and stain removers out there for just such an occasion - I've always had good luck with Quicksilver shampoo, just don't leave it in too long or her chrome will have a purple tinge to it... Cowboy Magic Green Spot Remover is wonderful, but expensive.
For a very small fun show, leaving her mane loose but clean/combed is fine, but at larger events her mane should be banded for Western or braided for English (either button or flat braids, whatever flatters her neck more). To be totally correct, her mane should be pulled to about 4 inches in length, but that's up to you and your preferences. Her bridle path should be clipped. For a western stock breed, the bridle path should be no longer than the length of the horse's ear flattened down. Hunter attire can go shorter, just a little wider than the crownpiece on the halter or bridle. Her legs should be clipped as well; curly feathering can be knocked off, and any ragged furries around her coronary band can be cleaned up. If she has a "billy goat beard" along her lower jaw, carefully knock that off as well. Whiskers are up to you - a lot of people won't enter the ring with whiskers, others won't dare clip them. Judges divide along similar lines IME. For a small fun show, I would just leave them. If you do clip her muzzle whiskers, DO NOT clip her long eye whiskers or her eyelashes. Her ears can be clipped by folding them shut and buzzing away the fuzzies that stick out. In bigger shows, some entries will shave out the entire ear, but that isn't necessary for a fun show, and won't be any fun for Lolly when the bugs are out. She can have a little hoof oil as well, either black or clear. The color is up to you, whatever flatters her most. Most Paints look better with clear - shiny black is just too much contrast for most of them.
Your 4-H leader and older members can help you out a lot with the preparations. Our club has a special show prep meeting early in the season, when the older and more experienced members show the newer ones how to clip and braid/band their horses and how to do the class. It's a blast, and it really helps the kids get started on the right track.
Ok, that got really long... sorry for the novel, but I hope that helps you a bit. Good luck, and enjoy your show experience!!