Translation:I drink black tea, and I also drink coffee.
Without getting into the weeds too much, the root word for both of those terms is のむ (飲む) and means "to drink"; this is the non-past version of the verb. Converting it into "masu-form" it becomes 「飲みます」; this is also "non-past" and means "to drink", but it's considered slightly more polite. On it's own 飲み is not the same thing as the verb "to drink". Edit: I originally said "shouldn't be the same" here, since I think 飲み can be short for "drinking" as in "to go drinking alcohol", but this is more of an exception.
The other example, 飲んで, is the て-form/te-form version of the base verb 飲む. Te-form has a few purposes. It can be used for making requests when you append ください to it. E.g. 「これを飲んでください」or "Please drink this". But it can also mean the "continuous" form of "to drink", i.e. "drinking". E.g. 「水（みず）を飲んでいます」or "I am drinking water".
You may want to look up "godan (五段) verbs" which are also called う-verbs, as well as "ichidan (一段) verbs" also called る-verbs. The root / dictionary form of ichidan verbs only ever end with a る, whereas godan verbs can end in u, ku, gu, su, tsu, nu, bu, mu, ru. All verbs in Japanese are either "ichidan verbs", "godan verbs", or are irregular (though there are only a handful of irregular verbs, like "to do" する and "to come" くる).
The difference is akin to comparing the following phrases: 'I drink tea, and also coffee.' and 'I drink tea and coffee.'.
I think that there is an assumption that people are either tea-drinkers or coffee drinkers. This phrasing could possibly emphasize that this person doesn't affiliate exclusively with either faction.
It really depends on the context and how the sentence is written. I would say that 'with' might sound more 'natural' to some people in some circumstances, but to say that it is better is debatable.
トムさんとご飯を食べました - I ate (the) meal with Tom.
トムとジェリーがご飯を食べました - Tom and Jerry ate (the) meal.
The first sentence could also mean that you ate 'Tom with rice', but that would be the less logical translation given context. You could also say 'Tom and I ate (the) meal', which would have the same meaning as the initial translation provided.
The second sentence could be translated as 'Tom ate (the) meal with Jerry'.
With/And usage is based on context and grammar.
トムとジェリーとご飯を食べました。 - 'I ate (the) meal with Tom and Jerry.' In this case, both 'and' and 'with' are used for と, as appropriate.