Let's make few things clear: as I mentioned before, you need to be ready to learn new stuff. But this does not mean that you should be afraid. I had no experience with VPS before, but with help of internet and its good people I had my LES machine running the way I wanted in short time. The best part was the amount of info I learnt in process.
Your first task is to go for LES website. For your own good, read it from beggining to end. If there is stuff that you do not understand, research it. There will probably be a lot of new stuff, but do not worry, (when writing this) I am running my VPS for less than a week and there is still stuff that I am not familiar with. This guide will cover all basic understanding you need to begin with.
So you studied LES website, right? Then let's talk about few things I think you need to consider at this stage.
LES VPS is unmanaged- based on OpenVZ and SolusVM. OpenVZ is virtualization technique, SolusVM is VPS management system "which allows you to seamlessly reinstall, reboot, shutdown, boot, change your hostname, get network information, get statistics regarding your server and much more". Imagine that VPS is your computer- unmanaged now means that you need to take care of everything: software install, update, user management, security, backup...
Let's check what we can get here. That's not so important for you, so go ahead and click on ORDER NOW. We are decided to get one, right?
Please, before signing up, consider visiting LES website with my affiliate link. It will help making this website better.
Now you see specs of VPS. What you do is research what those parameters mean (RAM, HDD vs SSD, amount of transfer, location). And choose one. How? Well, after you research those values, you should be able to make a decision, depending on what you want to do with your VPS.
In case you're wondering what you could do with 128MB, there is a nice post on reddit.
But back to topic. You should be done with choosing your location, and I guess, you've already clicked Order. I highly recommend carefully reading everything! Especially "Additional Required Information" part!
Take a minute to read "Additional Required Information" again. There is no need to hurry, so make sure you understand what you are doing. If you are not, there is a LES subforum where you can ask, so it might be not so bad idea to register on forums before buying.
There is some stuff you need to fill before checkout. Hostname is like name of you PC, so fill the name you would like to call your VPS; NS1 and NS2 Prefixes are not important now, fill whatever. Root Password should be self-explanatory. But in case you are not familiar with UNIXes, imagine Root as Administrator account of you VPS.
And for security's sake choose some strong password!
... or there's no way back, now.
To be honest, I do not remember all the specific details after checkout. Just be carefull to read all stuff (including emails!) and you'll be OK.
Reminder: Never checkout more than once, use proxy, TOR, or similar stuff. Everything you do is logged, and doing things mentioned before will get you banned. And be patient, the setup is not instant.
Talking about emails, you should receive 3 of them. Order Confirmation, Welcome (with client area's login information) and Customer Invoice. I know I am boring, but again, read them carefully.
And wait. Or go check your client zone where you can see your status under My Services. In case you got flagged as fraud but you are sure that you haven't done anything wrong, head to forums. There have been people successful with getting unbanned.