Learn English – is there any difference between “at first”, “initially” and “at the first time/ the first time”

They mean the same in those two examples, and both are grammatical, but I dislike as per. To me it sounds stilted and I can think of no occasions on which it is necessary to use it. That, however, is a personal view with which others may disagree.

As indicated by narx's answer to a similar question on ELU (English Language & Usage, ELL's "sister site" aimed primarily at linguists, etymologists, and serious English language enthusiasts), only little can be used as an unqualified noun to mean a small amount (where adjectival small needs an explicit noun, such as amount or quantity).

The other interesting point made about little there is that in...

1: I have a little experience teaching English.
2: I have little experience teaching English.

...the two sentences have almost opposite meanings - #1 implies you have at least some experience, whereas #2 implies very little (perhaps almost none). In fact, #1 is often used somewhat facetiously to imply you actually have a lot of whatever is being spoken of.

Apart from the above distinctions, there's also (as kiamlaluno points out) the fact that little often carries more "affectionate" connotations than small. And from BBC Learning English, little is more common in the sense not much when coupled with abstract nouns such as hope, chance, change, effect, use and point.

In most other contexts, there's little to choose between the two words. Use whichever you like.