I'm also going to go in order of when the band affected me.
3 Doors Down
The first rock band I ever listened to. My dad made me a mix CD of them when I was 5 or 6. Absolutely loved it.
One of my favorite movies when I was a young lad was The Scorpion King. I ended up getting it on DVD, and the music video to "I Stand Alone" was on there. I absolutely LOVED it and yet again, my dad made me a mix CD of them. They were my first introduction to hard rock/metal although it didn't really bridge me to anything. I still listened to softer rock like Aerosmith and Creed up until middle school.
I started really getting into Metallica when I was in 7th grade. I'd always listened to songs like Fade to Black and Nothing Else Matters but I started exploring their other stuff. I went through a straight year of basically listening to NOTHING but Metallica, new and old. I listened to Master of Puppets probably about 200 times throughout 2008. To this day they're in my top 3 favorite bands.
I ended up seeing this band live before actually listening to them (my friend dragged me to Mayhem 09 before I listened to much metal) but I had so much fun that I looked up a ton of their stuff. It's because of KSE that I was able to grow used to any type of screaming in my music because I still only listened to metal with clean vocals.
The first time I listened to Hatebreed I absolutely hated them. I didn't "get" hardcore music at the time, but over time they grew on me. They were the first band I ever regularly listened to that had a punk edge to them.
I listened to Crack The Skye for the first time on a long car ride a few years ago, and I've been in utter love with the band ever since. I didn't have much experience with progressive metal before that album, but it was a hell of a place to start. When I started exploring other albums like Leviathan and Blood Mountain I also discovered sludge/stoner metal, which I also love to this day. If it weren't for Mastodon I wouldn't have ever found some of my other favorite bands like Baroness or Kyuss. I probably would have never even further explored what is now my all time favorite band, Tool.
As I Lay Dying
I went through a sort of metal elitist phase at the beginning of high school, in which I denounced all things modern metalcore, but I eventually grew out of it and started exploring. I wound up discovering what is now one of my all time favorite bands. I was drawn in at first by the combination of heavy songwriting with generally catchy choruses, but it was the band's lyrical content that made me stay. There's something about Tim's lyrics that connects to me despite the fact I have a very different set of beliefs than him. AILD also bridged me to one of my favorite other bands, Parkway Drive.
That's only 7, but I can't think of 3 others. I could list 3 other bands I like but I can't think of any bands that bridged me to different kinds of music really.