Learn Russian: Здесь дождь.

Nouns ending in a soft sign (-ь) can be either masculine or feminine, so you usually have to memorize their gender as you learn the word. Дождь is a masculine noun. A helpful tip for beginners: many basic weather nouns in Russian, like снег (snow) and ветер (wind), are also masculine.

Why is there no verb for is or raining in this Russian sentence?

Russian does not use a filler word like it or a present-tense is to describe the weather. Furthermore, while English uses the continuous verb raining, Russian often just points out the noun дождь (rain). You are literally just saying Here rain.

I previously learned to say идёт дождь for it is raining. Can I use that here?

Yes! You can absolutely say Здесь идёт дождь (literally Here rain goes). Dropping the verb and just saying Здесь дождь is simply a shorter, very common alternative to describe the current condition in a specific place.

Both words end in a soft sign (-ь). How does that affect how I say them out loud?

The soft sign has no sound of its own, but it tells you to pronounce the consonant right before it softly. You do this by pressing the middle of your tongue against the roof of your mouth. Keep in mind that because дождь is at the very end of the sentence, its final softened consonant is devoiced, making it sound much more like a soft t than a d.