Underline adverb phrase and circle the word it modifies:

With only slight hesitation, Diane approached the microphone.
AP: With only slight hesitation
Word: approached

Peter quickly slipped under the gate.
AP: under the gate
word: slipped

Outside the theater a large crowd watched the arriving celebrities.
AP: Outside the theater
Word: watched

We sometimes drive far into the country.
AP: into the country
word: drive
I think that I get it now, but i was wondering if someone could check to see if i am doing these right?

An adverb is a word which modifies a verb, adjective, or another adverb. Circling and sentence diagramming is impossible in this format, as well as beyond the scope of a valid question. Since you already have the Adverb phrase (AP) and the verb in each case, I don't know what further help you need. The prepositional phrase in each context is used as an adverb, modifying the verb, is that your confusion? In the second sentence you also have an adverb quickly which is a direct modifier to the verb slipped, but it is not a phrase.