Major 3rds are always tricky to learn at first because they start incorporating a lot of sharps and flats into the mix. Hopefully this chart helps!
The biggest take away here should be that you need to pay really close attention to spelling.
Notice how all these are categorized into "groups", like the C and E group, and the G and B group.
For example, why can't you just spell E to Ab as a major 3rd? Well, as far as the actual note sound goes, Ab SOUNDS the same as G#, so why can't you use Ab to spell a major 3rd above E?
Well, let's count on our fingers to find out.
E is 1, F is 2, G is 3, and A is 4.
Having some kind of E go to some kind of A note means the interval will be some kind of 4th, not 3rd. Even though it sounds the same.
Now before you go and think music theory rules are being super picky here, just realize that in music it's literally the same thing as saying, "A bare lives in the woods." It may sound the same when you say it, but it’s the wrong spelling of the word “bear”. Make sense?
Anyway, hope that helps guys! Let me know if you have any questions!