To answer your specific question:
To quote from Leslie Lamport's "LaTeX: A Document Preparation System" (Leslie Lamport initially developed LaTeX):
The ten special characters
# $ % & ~ _ ^ \ { }
are used only in LaTeX commands.
...
Seven of those symbols can be produced by typing a \ in front of the corresponding character.
$ & % # _ { } are easy to produce.
In other words, to get { and } you must type \{
and \}
. This is because {
and }
mean something very special to LaTeX. They are used to delimit arguments. So with something like \emph{Hi}
, the emphasis starts at the {
and finishes at the }
(\emph
is a command which is used to emphasise text; the standard behaviour is for it to italicise it). In other words, everything enclosed within the braces is emphasised.
As such, if you want to print the symbols { and }, you can't just type the plain symbol. Instead you must "escape" it. The \
is the escape character and it basically tells LaTeX to pay attention. emph
is just a word like any other to LaTeX. But \emph
is a command to LaTeX. What escaping a {
does is cause LaTeX to print a { character.
So to get {2,4,6,8...}, all you need to do is type:
$\{2, 4, 6, 8\dots\}$

To get {x|x is positive and even}, you can do:
$\{x \mid x \text{ is positive and even}\}$

As raised in the comments - this relies on the amsmath
package. Instead, you can do:
$\{x \mid x \mbox{ is positive and even}\}$
For most purposes, this has the same effect, but there are minor differences:
Difference between various methods for producing text in math mode
For really nice spacing, follow Hagen von Eitzen's advice and use:
$\{\,x \mid x \text{ is positive and even}\,\}$

As you can see, \,
inserts a thin space. There are other commands for inserting space in LaTeX. However, \,
is one of only two that can be used outside of math-mode.
But, really, it's important that you gain an understanding of the basic operating principles of LaTeX before you go further and start worrying about how to do specific things; otherwise you're gonna be completely at sea and everything's gonna seem like black magic and nonsense. So I really recommend you have a little look through some basic introductions before diving right in there.
However I learned by doing - I learned by trying it out, I learned by using LaTeX and looking things up whenever I wanted to do something I didn't yet know how to do. That's a very fulfilling, very enjoyable and very practical way to learn - although there are massive holes in my knowledge and, because my approach was very light on theory, I'm only just starting to learn about bad habits I've developed and principles I may have overlooked.
But still, as good as it is to dive in there and get your hands dirty, you have to, have to get a basic idea of what LaTeX is and how it works or you'll probably continue to have a pretty miserable time. It's no wonder you don't like it and everything seems arcane and difficult and a complete pain in the posterior to you!
latex3
so I will edit the tags in a moment. However, please try$\{thing 1, thing 2, thing 3\}$
. – cfr Jul 1 '15 at 3:26