This is not an AmE or BrE issue at all.

In English, we say:

**I like it here.** or **I like it there.** to mean: I like being or living in some place. 

If you are in the place (city, town, area, region, country, etc.), you say: I like it here. If you are not, you say: I like it there.

This is basically idiomatic. **The it stands for**: living or being in a place.

Also, yes, in English, we say: I visited a place [city, town, region, country) and also say,for example, for that (tourism): I went to [some place] last year. 

There is nothing wrong with saying "visit a place" in English and again, there is no difference between AmE and BrE.  

[the verb like has to be followed by an object (verb or object) unless it is being used with the function word **to**.]