In the International Phonetic Alphabet (IPA), finite is /faɪnaɪt/ and infinite is /ɪnfənɪt/.
In case you are unfamiliar with the IPA, "finite" rhymes with "skylight." Both of the vowel sounds are the "long i," as you put it.
You might hear "infinite" pronounced in two slightly different ways. The first way rhymes closely with "run it" (in-fuh-nit) and the second way rhymes more closely with "win it" (in-fin-it). The stress in either pronunciation will fall on the first syllable.
If you are (or are willing to become) familiar with the IPA and its pronunciations, you can find a helpful tool here that transcribes most English words into their IPA form.