How do you say after (insert verb here) in Japanese?
like after studying, after drinking? would after studying be benkyotte kara? (べんきょうってから）
There are many ways to say "after ...ing" in japanese. There is no one to one translation, since you can use different words as "after" depending one the overall meaning of the sentence.
One of the most common translations and one of the first one learns would be:
This shouldn't be confused with the reason-giving から which is not used with the て-form of a verb.
家に 帰ってから すぐに食べました。(ieni kaettekara sugunitabemashita.) "After returning home, I ate immediately."
Note. I believe it is better to say that there is one から. The distinction between the "after"-から and the reason-giving-から is there for learning purposes. If one where to give one translation for から that would be "from". Eg 「日本から」(from Japan). So, in the above context of time, the exact translation would be "from (the time I have) returned home, I ate.", meaning that "I started eating from (as origin) my having returned home". For more info on から and to fully get the gist of it, read the excellent IMABI
Another usual translation is:
後 means "behind" when you use it in a spatial context (and it is then pronounced as 「うしろ」) or "after" when you are speaking about time (and it is then pronounced as 「あと」. So, in a way, the concept of "after" for the Japanese is similar to that of "behind", while "before" is similar to "in front". This is particularly interesting in 「午前」 and 「午後」（ごぜん and ごご）that mean "am" and "pm" respectively. You could also use that formation in other words, eg 「今後」 which means "from now on", literally "after now".
家に 帰ったあとで、すぐに 食べました。(ieni kaettaatode, suguni tabemashita.) "After returning home, I ate immediately."
Note that 「、」 was not traditionally used in Japanese and it is written there mainly to help foreigners.
Another way to say after is by using just the て-form of the verb.
家に 帰って、食べました。(ieni kaette, tabemashita) "After I returned home, I ate"
Note: The thing with て-form is there is no exact match for this grammatical phenomenon in English. If I were to choose I would say that it is closer to the past participle. But that is just an, at times, convenient translation. If I were to literally translate this sentence then, it would be "At home returned, I ate." In any case, one has to just accept things as the are at some point!
Also, note that the て-form can be used to give reasoning.
You can also use the -ます form of a verb without the ます ending, to denote the succession of actions.
Note: This is more like a listing of the things you did. The ます form of the verb without ます is a strange artifact as in many cases it is used as a noun. For instance, 帰り in the example sentence means "the returning, the coming back". So, again, if you want to translate exactly the example in English, you will have some trouble. If you feel it helps translate it as the present participle.
In more advanced levels, one learns of many other expressions. Those expressions are used depending on context and, while they might be translated as "after", the meaning takes a special nuance. For instance：
-たら： this is one of the few ways to create a hypothesis. But, as in English, hypothesis might have a suspicion of time. Eg 「家に 帰ったら、電話をかけて。」(ieni kaettara, denwawokakete) "If/When you arrive home, call me."
-とたん： it means "exactly when", "just". Eg 「寝たとたん、眠りました。」(netatotan,nemurimashita.) "Just as I lied to sleep, I fall asleep.", This can be translated as "exactly after".
-次第（しだい）： it also means "exactly when", "just", but the use cases are different. Literally, it means "the next in line", so it is frequently used to request, suggest, ... actions. Eg 「家に 帰り次第、電話をかけてください。」(ieni kaerishidai, denwawokaketekudasai.) "First thing to do when you return home, please call me.) Again, it might be translated as "immediately after you arrive home,...".
-上で： it is a more polite version of -てから. Here, again, we notice the use of a spatial adjective to express time.
-上は： it is better translated as "since" or "given that", eg 「試験を受けることにした上は、勉強するしかない。」(shikenwoukerukotonishitaueha, benkyouusurushikanai.) "Since I decided to take the exam, there is nothing else to do but study". Here, "since" is more like "because" but it also contains the meaning of "after".
以上： quite similar to 上は, but it rather gives reason that expresses time.
The list could be going on, but the more one adds, the more they deviate from the meaning of 'after'. All in all, if you are a beginner, stick to what the books say. "-tekara means after", "taatode means after" and so on. As you become more comfortable and acquainted with Japanese, try to sense those words. Listen to how Japanese people use them, their emotion and so on. Not everything can be translated 100%. Also, some words are more usually written than said and vice versa. The above is how I understand things.