I read on another site that when a character is thinking something, what they're thinking shouldn't be italic since it isn't the right use of it, but I do this because it's easy for me to remember what's being said out loud and what someone is thinking. If I have been doing the wrong thing, how should I show someone is thinking?
There isn't much text that is thinking maybe ten lines if put all together, so it's save to say the reader won't be overwhelmed by italic text.
The only time I've used italics in my writing is to show my main characters dreams.
Like you, I don't know if this is correct, and I only do it to show that it is different to the main body. Perhaps when and if I hand it over to a literary editor they may suggest something different, but for now it works for me.
I think I'd echo Jonathan's thoughts and suggest using inverted commas around the text along with 'he/she thought/wondered' as that also seems perfectly acceptable.
I think (sorry!) there's no real right and wrong - the crucial thing is to be consistent.
The problem may be that if your character thinks a lot you can end up with blocks of italicised text which are...annoying, if you like, to read.
Personally I use italics to emphasise single words in a character's speech or for non-English phrases. And for ship names, which is an odd literary convention.
If you don't want to italicise thoughts, 'he/she thought/wondered' etc works okay. I use single person POV so I can often just write a character's thought without marking it as such because it's immediately obvious from the preceding narrative.