You're stuck with the ice surface, but I'm going to help you choose a boot and a blade that will keep you from being able to blame them (sorry).
First a word about brands. Brand is important in figure skates, even low-end figure skates. I've never seen a no-name brand that I could recommend. (This includes the ones they sell at L.L.Bean. These are the worst skates on the market.) The brands to look for in the States are Reidell, Jackson, SPTeri, Klingbeil, and Graff (help me out and let readers know 1-who I'm forgetting, and 2-any Canadian, UK, or Aussie brands that I don't know about). I know that the first three make recreational-level skates. Don't worry about blades yet (more below).
Boot and blade set. A lot of adults swear by what I call "comfort skates"-- those recreational skates that look like sneakers. If you're a brand new skater these are probably your best bet, although some adults (cough*me*cough) find that they don't feel like they're giving you enough support. Try a few brands--Reidell and Jackson brands are fairly easy to find.
For children 6 or younger a beginner recreational skate like Jackson SofTecs are the only way to go. Standard figure skates are very uncomfortable for a child used to sneakers 2 sizes too big. Children this age have an easy time switching to standard skates after a year of skating, and it's simply not worth the barrier of uncomfortable skates. Older kids I would put straight into standard beginner skates like a Jackson "Mystique." Kids over the age of 6 are going to understand the concept of fit better, and also tend to learn faster, so you want them in a skate they can wear through several levels of skills.
Higher level boots are stiffer than beginner boots, because jumps require more support. They will tend to have a higher shaft, although this is changing with a lot of high level skaters choosing a cut-down back so they can point their toes. And this is the single most misunderstood aspect of skates. Skates are padded at the sides of the ankle to support and minimize lateral movement. You want to be able to bend your ankle forward, very very deeply. High level skates actually put in things like "scallops"--little cut outs, sometimes as many as three, so that you can bend your ankle in a stiff skate. The thing for recreational skaters to remember is that it takes a lot of muscle power to "break" a skate like that-- i.e. to get the ankle to bend properly. Put a 60 or 80 pound Beta-level child in a high-level boot and they will simply not have the weight or strength needed to bend their ankles properly, which certainly affects how well they skate, how comfortable the skates are, and can also lead to injury.
Let your budget guide you--you don't have to spend a thousand dollars on skates, although you can; there are excellent choices around $200. Think of it as comparable to the cost of a decent bike.
Factors in blades are: toe pick, hollow, and radius.
Get a toe pick appropriate to the jumps you are doing. No jumps? Beginner blade with a small, high toe pick. Beginner jump? Beginner blade like an MK Pro, Wilson Majestic or the blade that comes on a skate like the Jackson Freestyle. Again, talk to a skate technician and be honest about your skater's level and ambition. Don't put a class-only skater in a $400 blade.
The hollow is the curved area between the edges. You can get it in different radii, from 3/8" (very deep/sharp) to 1" (very flat/shallow). Unless you're a really really serious skater, don't worry about it. I mean it. However, the technician who sharpens your skates does need to understand hollow. Never never never get your skates sharpened by anyone except a certified technician. Don't go to the local bike shop that bought a machine because they thought it would be a good sideline. If the nearest sharpener is farther than you want to travel, ask around your rink--often the high level coaches will have their own machine and be trained and skillful at this, and will not charge any more than the skate shop. And seriously, you do this twice a year, maybe 4 times if your kid skates a lot (a skater can get 40 to 60 hours of skating out of a sharpening). It is ridiculous to risk ruining a $100 blade to save six bucks or to avoid a half-hour drive.
Again, don't worry about it until you're ready for your first pair of expensive blades. The rocker is the radius of the curve that the blade is cut on; i.e. a blade is basically a little piece of a circle of either 7 feet or 8 feet. An 8-foot rocker is a little flatter than a 7-foot rocker. Beginners tend to like the 7 foot one better, although it really doesn't matter which radius you use, at any level. You should know the "profile" of your skate (basically the silhouette), but the specs don't matter that much--they all work, and the depth of the hollow and radius of the blade are entirely personal preference. Again, your skate technician needs to be able to tell which radius your blade uses-for instance, hockey blades use a very large radius, with a sharp curve at the toe and heel. If your technician doesn't know this, and doesn't reset the machine for figure skates, you're screwed (this happened to a large group of the rental skates at my rink--they sent the figure skates to a technician that gave them all a hockey grind--took the back of every blade right off. Sadly, when I pointed this out to them, they had no idea what I was talking about.)
Never put a beginner (through Gamma) in a freestyle blade. A beginner blade has a toe rack with all the picks roughly the same size, and starting high up the rocker (curve of the blade). A freestyle blade will have a larger top and bottom pick, and the whole rack is farther down the rocker, i.e. closer to the ice. The time to switch to a freestyle blade is whenever the child outgrows the pair she's wearing around Delta or low freestyle. Freestyle skates come in both boot-blade sets, and separate boots and blades.
If you buy separate boots and blades, the blades can often, but not always, be used through a couple of size changes in the boots. Talk to a skate technician at a specialty skate shop to determine whether your old blades are appropriate on your new boots.
If you're a beginner I would simply not buy used skates from your neighbor, friend, Play-It-Again Sports, or that nice mom at the rink. You don't know enough about it. I cringe every time I see some 6 year old PreAlpha skater in a freestyle blade and heavy boot. It's like putting a new rider on an Arabian dressage horse instead of a pony. Not safe, and not fun. If you get used skates, get them from the specialty skate shop
A lot of information. How do you choose?
You can do your own research--there's tons of advice on the web. But a better way to do it is to ask a coach or a specialty skate shop. Give them your budget and your skating level, and they'll be able to steer you to a good skate. A lot of coaches will simply recommend the boot and blade that they like; this is fine. As you progress as a skater, you'll get more knowledgeable and start being able to make your own choices--my daughter skated in SPTeri's for years because that's what I wear. Then one day, about the age of 14, she decided, more or less on her own, to try another brand, and now she's a dyed-in-the-wool Reidell advocate.
A lot of coaches, even high level coaches, do not educate themselves about skates, but their advice to just buy the skate they like is still probably fine. The Professional Skaters Association, strangely, requires knowledge of boots and blades only from coaches taking the Group Instruction rating; it's not part of the Free Skating, Dance, or Moves exam, which seems utterly bizarre to me, but of course, they don't ask me.
Long story short? (Or as they say now in web parlance tl;dr-too long didn't read) It's a good idea to understand all the techinical aspects of boots and blades, but not necessary to end up with a decent set appropriate to your level. Talk to your coach or a specialty shop and you'll end up with a decent boot.