Can there be more than one verb in a sentence, such as one linking verb and one action verb?
A sentence requires one verb.
That one verb may have auxillary verbs, the auxillary verbs being the "-ed" or "-ing" form of the word, as well as forms of "be" or "have". Unless you really need to drill down to the word level, it's best to think of the sentence as containing one verb that is expressed with multiple words.
I am going to the store. <- "am going" is the verb.
You can have one subject and multiple verbs in a sentence if you link them with a conjunction. Repeating the first auxillary verb is usually not necessary, but can be done for emphasis.
I am running and chewing gum at the same time.
I am running and am chewing gum at the same time.
I love and hate that movie. Let me explain.
Words that are "-ed" and "-ing" verb forms, as well as infinitives, are not always verbs in a sentence. They can act as nouns, or as part of adjective/adverb phrases.
To keep watch is necessary for security.
"To keep" here is equivalent to a noun.
Walking is necessary to keep oneself in shape.
"Walking" here is a noun.
The following answer is the first, slightly modified part of one I wrote in response to a very similar question at English Language & Usage:
Theoretically, it is possible to have an infinite numbers of verbs in a single sentence. In actual usage, even simple sentences frequently contain more than one "main verb" in what is called a compound predicate.
Example: I closed my eyes and counted to ten. (Two verbs)
A sentence can contain additional verbs that are part of relative clauses, subordinate clauses, etc.
(subordinate clause) When I looked outside,
(independent clause) I saw the terrible damage
(relative clause) that the storm caused. (Three verbs total)
Furthermore, a sentence can have several modal/helping verbs working together: I must have been living there for six months.