You probably learned early on that 和 (hé) is used as an "and" word to connect nouns and noun phrases, but not sentences or clauses. This rule is fine for a while but eventually you will begin encountering what appears to be non-nouns linked by 和. This article addresses how you may see 和 used in other ways.
It's worth mentioning that none of these uses of 和 needs to be "learned." You could do just fine only using 和 to connect nouns and noun phrases, and native speakers may prefer not to use 和 in some of the examples sentences given below. Still, uses like these do exist.
Rather than give absolute rules, it's more helpful to give trends, because it's very difficult to provide a rule for using 和 with non-nouns which is never broken. There are just too many gray areas.
In most cases, the following trends apply when 和 is used to connect verbs:
The two verbs are very similar in nature. This could include characteristics such as level of formality, structure, and scope. Often you'll find that the verbs even overlap a little in meaning or are frequently used together. Thus "唱歌和跳舞" is natural (both are separable verbs, equally informal, frequently used together), whereas "玩游戏和跑" (the structure of the two is very different) is quite awkward.
Both verbs share the same object (e.g. "管理和运营着家公司") or are both modified by the same adverbial phrase (e.g. "进一步地了解和交流").
Normally (but not always) both verbs are disyllabic (two-syllable verbs), such as 维护, 修理.
In most cases, at least one of the following trends apply when 和 is used to connect adjectives:
The two adjectives do not act as the predicate of the sentence, but instead act as an attributive (coming before a noun), object (coming after a verb), or other sentence element. There will be some kind of modification before the adjectives; they won't just be used "bare"
Don't forget that in addition to 和, there are all kinds of other ways to express the "and" meaning, such as the words 并, 并且, and 还有, or the patterns 又⋯⋯又⋯⋯ or 既⋯⋯又⋯⋯.
The following sentences don't follow the trends laid out above, and are thus not good sentences.