I think you are actually talking about three different patterns. In particular, whether there is に between the two verbs is very important.
1. masu-stem + に + movement-verb
取りに行く, 遊びに来る, 寝に戻る, and so on. The first verb can be almost anything. The second verb is a "movement verb" (行く, 来る, 参る, 戻る, 訪れる, etc). The first verb before に expresses the purpose of the movement. The に is very important. See: Is it true that only movement verbs can take [V-stem]に to express a purpose?
2. Syntactic compound verbs (masu-stem + verb
)
やり直す, 見始める, 食べまくる, and so on. The first verb can be anything. The second verb is one of the 30-ish verbs listed in the "Syntactic and Lexical Compound Verbs" section of Compound Verb Lexicon. The second verb adds some meaning (e.g, "re-", "start") to the first verb.
3. Lexical compound verbs (masu-stem + verb
)
There are literally thousands of compound verbs where the combinations of two verbs have fixed meanings. Some are straightforward enough (e.g. 飛び出す, 立ち上がる), but many have completely unpredictable meanings (e.g., 差し支える, 見込む). The linked lexicon is for listing this type of verbs.
The first two categories are easy, but lexical compound verbs are the toughest, and you'll have to remember each combination one by one.