The correct sentence was said to be 'Mein Bruder spielt manchmal Klavier,' so it would literally translate to 'My brother plays sometimes piano.'
However, when another sentence goes, 'Ich koche oft,' it translates to 'I cook often.'
I know German doesn't have the same rules as English, but is there a general rule I can memorize to correct myself while making these mistakes?
One key difference between English and German is the position and order of verbs. Looking at simple sentences obscures the fundamental differences; the following pair is, after all, almost identical.
Solutions can1 be2 found3.
Lösungen können1 gefunden3 werden2.
However, the structural differences can easily be made visible.
(because) solutions can1 be2 found3 quickly by this simple algorithm
Lösungen können1 von diesem einfachen Algorithmus schnell gefunden3 werden2
weil Lösungen von diesem einfachen Algorithmus schnell gefunden3 werden2 können1
It is the subordinate clause of the last example that reveals the base position of German verbs: at the right edge of the clause, their order reversed (321) compared to the English equivalent (123).
Klavier in Klavier spielen isn't a completely normal object; the phrase as a whole refers to the habit of playing the instrument, not to playing on a specific instance of it or a specific act of playing. The easiest way to think of such combinations is that they behave almost like a new verb, and verbs (and their parts and objects "close to the verb") are positioned at the right edge of the clause. That's the reason why other stuff (such as manchmal in your example) has to go in front of it.
weil die beiden früher jeden Abend zusammen im Keller Klavier gespielt haben
because the two of them used to play the piano in the basement together every evening
As soon as you replace Klavier by a normal object, other orders become available. (As has been pointed out, German word order "in the middle" isn't fixed.)
weil die beiden dieses Spiel früher jeden Abend zusammen im Keller gespielt haben
German word order is very flexible compared to English, so your version isn't really wrong so much as odd sounding. Adverbs of time, which would include "manchmal" and "oft", tend to be placed early in a sentence. One possible reason to put "manchmal" at the end would be if everyone already knows your brother plays the piano, but there is some confusion about how often and that he plays and "manchmal" is the new piece of information. But if such a situation did arise then it seems likely that you'd move "manchmal" right to the front: "Manchmal spielt mein Bruder Klavier." So putting it at the end would be very unusual.
In English, adverb placement is fairly flexible as well, but putting one between the verb and object isn't allowed, so "My brother plays sometimes piano" sounds wrong. (To my ears it sounds like the speaker has a German accent.) There is no such rule in German and putting an adverb, especially an adverb of time, between the verb and object is common.
German word order has few "rules" as such, but there are some guidelines you should follow as a learner. These are commonly taught in textbooks, time-manner-place, pronouns before nouns, etc. Since they are really more guidelines than rules, so you may hear them "violated" by native speakers. But sticking to the guidelines ensures that your sentences are at least not wrong or strange sounding. So as a learner you're better off sticking with the "default" order: "Mein Bruder spielt manchmal Klavier."
Situational adverbials, in your case those expressing repetition, are usually positioned in front of the object, after the time, and in front of the place.
Since it more precisely depends on a whole bunch of variables, this is an oversimplified answer but true for many sentences.
To emphasize a point made RDBury:
You might hear the sentence with manchmal at the end, but it would be correctly written as:
Mein Bruder spielt Klavier - manchmal.
In that case the speaker became aware after Klavier, that the sentence sounded like the brother practised often or at least regularly and wanted to correct that impression without repeating the re-arranged version of the sentence. So the dash represents a small delay in speech.
