Enough chit-chat, let's get on with it.
Actually, before we get to "no mames", Let's talk about the verb mamar, meaning to suck. However, like most slang, it's usage goes well beyond its original meaning.
Let;s start with the word mamadas. Mamadas could be translated as lies, except that mamadas is one of those words classified as not for polite company. So with that in mind, I would translate mamadas as something closer to bullshit.
No me digas tus mamadas
Don't tell me any of your bullshit
You could also substitute pendejadas for mamadas. It's not quite as strong, but still not polite. I'd translate pendejadas as dumbshit.
No me digas tus pendejadas
Don't tell me any of your dumb shit
Next we come to mamada. Notice that we dropped the 's'. That small change in spelling makes a big change in meaning.
A mamada is a blow job. I usually like to use pictures to help drive the meaning home, but I suspect that would be a little too much in this case. Besides, I don't think much more of an explanation is needed, so I'll just give you a few examples:
Quiero que me hagas una mamada
I want you to give me a blow job
A mamazo is also a blow job, but not just any old blow job, it's a really great blow job.
Esa viejta me acabe de dar la mejor mamazo de mi vida
That old lady just gave me the best blowjob of my life
OK, now it's time to move on to "no mames".
"No mames" is one of those catch-all expressions that can mean damn near anything depending on the context. Translation is a tricky thing, so I'm reluctant to say that "no mames" directly translates to one particular word or phrase in English. The key here is that you need to understand the spirit of the phrase, which by the way in no means should be used in polite company.
Don't worry, "no mames" is something you'll quickly get the hang of, you just need a few examples.
Hoy me despidieron.
They fired me today.
¡No mames! ¿Porqué?
No fucking way! Why?
Págame mi dinero, güey.
Pay me my money dude.
¡No mames, güey! Te pago mañana.
Don't fucking worry dude, I’ll pay you tomorrow.
Oye mano, ¡gané la lotería!
Dude, I won the lottery!
No mames. ¿En serio?
No shit? Seriously?
Ever have someone who just won't stop bothering you? Just tell them...
¡No mames! Dejame en paz
Stop fucking bugging me, leave me alone
Depending on the situation, you could also translate the above as "Stop fucking around".
The truth is "no mames" is something you just have to experience and you'll figure it out quickly enough. And with that said, I'll leave you with a video.
You won't hear all the uses of "no mames" in this video, but it'll get you off to a good start and hopefully you'll get a few laughs out of it as well. I suspect that even if you didn't have my recommendations on how to translate "no mames", you'd come up with a translation on your own.
Hope you enjoyed the video. Be on the look out for more uses of "no mames". A simple internet search will give you more than enough material to make you an expert on the subject.