Were has no contraction. It stands alone, pronounced /wɜːr/ in UK English and /wɝːr/ in US English. In fast, unstressed speech it often relaxes to a quick /wər/, so you may hear it that way mid-sentence even though the fuller form is the one to aim for when speaking carefully.
We're is the short form of we are, sounding like we plus a quick r. We're late for the bus. The apostrophe marks the missing a of are.
We're here, Diane. The contraction keeps the /wɪr/ sound of we plus a short r glide, and the apostrophe only marks the missing a of are.
We were tired, so we're resting now. Say the first word /wɜːr/ with a longer vowel, the second /wɪr/ quicker, and the spelling choice follows the meaning.
Where the spelling trap sits
The trap is mostly on paper, not in the ear: /wɜːr/ and /wɪr/ are distinct vowels, but the e and r spellings get swapped when typing fast. Keep the apostrophe in we're; a plain were there changes the sentence.
In the strong UK form /wɜːr/ the vowel is long, and the e is the vowel, with r following it. For the printed forms were and we're, no letter is silent; the one missing piece is invisible, since the apostrophe in we're stands in for the omitted a of are, so the printed letters that are there all sound.
Weak forms exist too. Unstressed were can reduce to /wə(r)/ in British English and /wər/ in American English. That reduction is about stress, not spelling, so it does not license dropping the apostrophe from we're.