Learn Dutch: Het weer was vreselijk. - The weather was awful.

You would place it directly after the verb was. To say 'The weather was especially awful', you would say: Het weer was vooral vreselijk. The same rule applies to zelfs (even).

Does the word weer also mean 'again'?

Yes, they are completely identical in spelling and pronunciation! Het weer is a noun meaning 'the weather', while weer is an adverb meaning 'again'. You can easily tell it is the noun here because it follows the definite article het.

Why doesn't the adjective vreselijk get an -e added to the end?

Because it is used here to describe the subject after a linking verb (was). Dutch adjectives only get an -e ending when they are placed directly in front of the noun they describe. If you changed the phrase to 'the awful weather', you would add it: het vreselijke weer.

How do I pronounce the -lijk ending in vreselijk?

Even though it is spelled with the Dutch ij, the suffix -lijk is pronounced like 'luk' with a weak, short 'u' sound (a schwa). You will never pronounce the full 'ij' vowel sound in this specific suffix.