出来る is a verb meaning "Can/be able to".
Which ways of attaching a verb to it is correct?
i.e. 覚えて出来ます。
i.e. 覚えるのが出来ます。
I feel, like both should be correct, but I'm just still just a half year learning japanese, so sorry for trouble. Are there any other ways to say "can"?
If you are using ができます to express the ability to do a verb, then it has to be V(dict. f.)＋こと+ができます。You cannot use の to nominalise the verb in this case.
The main other way to say "can" for a verb is the potential verb form, which I assume you are yet to encounter :) -- so here is a summary of how to form it! (See also the entry in Tae Kim's Guide to Japanese Grammar, who probably does a better job than I... here).
For Group 1/'う' verbs (e.g. 行く、飲む), you change the final sound to the え-line, and add る e.g. 行く→行ける (can go)、飲む→飲める (can drink). These then conjugate like Group 2/'る' verbs.
For Group 2/'る' verbs (e.g. 食べる、信じる (a Gr. 2 exception)), you cut off the る, and add られる. e.g. 食べる→食べられる (can eat), 信じられる (can believe). These are the official forms, but there is also a slang-ier contracted form for the Group 2 verbs, where you drop the ら e.g. 食べれる, in case you encounter these. But you probably don't have to worry about that for now...
For Group 3 verbs, you have 来る→来られる (can come), and する→できる (can do, as in the Vることができる construction).
This post explains that there is some nuance between the type of "can" that each of these constructions expresses (i.e. ことができる perhaps implies a technical ability, while the potential form of a verb has a broader meaning): https://japanese.stackexchange.com/a/2719/33435
There are a few other ways you might express potential (e.g. using the noun for potential 可能), but the two ways above are the most important means of doing so.
Hope that helps, and good luck on your Japanese learning journey!
