Auch mal is one of those little idioms that are used a lot in spoken language and can have dozens if not thousands of quite different meanings depending on context. Just think of the many uses of just, like or once in English.
Etymologically, auch mal is short for auch einmal, so literally it translates to also once. (Einmal, literally one time, has been shortened to mal, literally ... time.) Perhaps the most basic situation in which you can use it is when addressing someone who has been monopolising a new plaything:
Lass mich auch mal dran! - Let me have a go at it too, for once!
In this case both words have separate, clearly distinguishable meanings:
Lass mich dran! - Let me have a go at it!
Lass mich auch dran! - Let me have a go at it, too!
Lass mich mal dran! - Let me have a go at it, for once!
But note that the translation of mal as for once is overly literal and very misleading. Mal is a casualness marker - one of German's rich supply of discourse particles. Some of them, such as eigentlich, have rough equivalents in English (in this case actually). But mal doesn't, which will make it a relatively strange new concept for English speakers trying to learn German. When translating from German to English, mal as a casualness marker sometimes has to stay completely untranslated, and sometimes it is best translated by switching to a casual English idiom instead of the straightforward formulation used in the original:
Schreib mir einen Brief! - Write me a letter!
Schreib mir mal einen Brief! - Drop me a few lines!
Some languages use diminutives of nouns instead. Usually this works neither in German nor in English, but you can create a similar effect in English by adding an appropriate adjective to mark the casual aspect:
Schreib mir mal einen Brief! - Write me a quick letter!
Now to your example sentences.
Ich gehe auch mal zu diesem Café. [Note that it's much less acceptable in German than in American English to drop accents on French words. German keyboards even have a key for producing them.]
This sentence is a bit strange, since in German you don't normally go 'to' a café (meaning you stay outside), but you go 'in[to]' it. So let's fix that first:
Ich gehe auch mal in dieses Café.
This is a sentence that might well occur in a conversation, but taken out of context it is not clear what it is supposed to mean. I can think of two interpretations:
Ich gehe auch immer wieder mal in dieses Café. - I, too, go to this café now and then.
Ich gehe dann auch mal in dieses Café. - So I'll go to this café too. [There is only a slight hint at the casualness marker in the translation: the contraction of I will to I'll.]
The second example sentence is outright ungrammatical because it mixes two constructions (and misspells one of the words involved):
Schön, dich auch mal zu sehen!
Schön, dass ich dich auch mal sehe!
In this case, mal works only superficially as a casualness marker. Its original meaning [for] once shines through under the veneer of geniality, so it's likely intended (and will be understood) like the somewhat accusatory:
Nice to see you too, for once!
(German and English grammar are still similar enough to translate the ungrammatical blend "Schön, dass dich auch mal zu sehen!" back to English: "Nice that to see you too, for once!" This translation even works when taking the original das instead of dass seriously, as both can be translated by that.)