Translation:Let us join the Esperanto club in our area.
I believe that "(la) esperanta klubo" would mean something like "the club of hoping ones" or "the club of these who hope"... But I believe that it isn't so necessary to use the capital letter because I don't know how about you but I haven't heard about any "club of those who hope" ever, so I believe everybody would understand that I am writing about Esperanto related club or any other thing...
I assume that aligxu is alio (other) + -igx (become, intransitive) + '-u' (imperative), which would be roughly become something else.
Is this how aligxu translates to "join"?
Edit: It seems that ali- is the base for some pseudo-correlatives and that aligi (to add) and aligxi (to join [one's self])) are built upon that. So it seems it's not quite accurate to say that aligxu == ali[o] + -igx + -u, and neither is it turning any transitive verb into an intransitive verb.
Then again, I could be completely wrong.
A note I read in Montagu Butler's fine "Esperanto-English dictionary" (my first go to for English meanings of E-aj words) says: "A proposed correlative series alial … is inadmissible (alia, alie, having other meanings). Nevertheless, alies (col)=someone else's, though unorthodox, is useful." Otherwise everything you say in your edit seems pretty much correct. (except that aliĝi breaks into al+iĝ+i "To become towards.")
The ellipsis (…) exists in the text I copied.
I saw that (about alies) about a week or so after I'd written the above. I found it while looking for something else and could not, at the time, remember where I had written this. I thoroughly agree with the PIV about aliel, I can see some occasions when alies could be useful. In fact; in my webcomic ( http://www.littlelevers.com/angxeloj ) I have a character who is supposed to be under-educated, and I keep looking for a suitable occasion for him to use it. It seems the kind of thing he'd say.
It's really hard (though not impossible!) to express the concept of a character with a specific dialect in Esperanto, since people generally strive so well to meet an international norm/standard. I think it's fascinating to use 'alies' as a way to indicate a less educated character... like leaving out the first r in 'library' i suppose!
FWIW, the reason 'alies' and 'aliel' are marked 'evitinda' in PIV is because 'ali-' isn't part of the table of correlatives, and therefore can't actually take the '-es, -al' endings. Though sometimes I imagine that somewhere there's a very close parallel universe where the only difference is that 'ali-' (other) is included in the table of correlatives, and elephants have prehensile trunks and tails...
But I digress... the word in question was aliĝi, from which we have now afielded far...
They could just be separate words though, not ali with endings es and al. Either way, I reckon PIV recommends they be avoided for they may cause confusion and are simply not widely used, I suppose. One might recognize the es ending of alies as a correlative, thence via extrapolation conclude that alie means ‘elsewhere’, which it does not. (Then again, there are plenty of words that may have a similar problem: take ekvatoro (equator). When I first came across this word, I recognized the ek prefix and was puzzled when I couldn't find the word *vator/i in my dictionary!)