Alright, here's the lowdown on what I'm looking for:

I have absolutely no idea what I'm doing when it comes to music.

So far, I've found myself drawn to random songs that I've heard here and there, and they mostly tend to be punk-rock or along that line, like Green Day, Fall Out Boy, etc...

Can anyone suggest any other bands like this? Or maybe just anything you think may interest me. I want to be a rounded individual.

Also, no screamo or really hardcore rock, please. The lyrics are just as important as the instrumentals, and I'd rather be able to hear the lyrics.

Incubus(1), The Killers(2), and Red Hot Chili Peppers(3) are some.

Like Anna Molly(1), When You Were Young(2), and Dani California(3).

Look up a song called "Let the Drummer Kick"

I've taking a liking to that song.

Please ignore what Tif said about Red Hot Chili Peppers... No offense to her, or the band, but Dani California just isn't anywhere near what they really are. It's just turned into a big pop hit because Rock Band brought it back. If you want the "real" Red Hot Chili Peppers experience so to speak, check out the albums "Blood Sugar Sex Magik" "Mothers Milk" and "Californication." Stadium Arcadium wasn't a bad album, it's just that it was a tamed version of Californication.

Though if you want really good lyrics and instrumentals, just listen to anything by Rush. Seriously, their music is incredibly orchestrated and just really really really well written to put it simply. Then Neil Peart writes incredible lyrics. He's such a good writer that he's even become a best selling author (though his main job is drumming and writing lyrics for Rush).

I like "Under the Bridge" by Red Hot Chili Peppers ^^

Lets see (you might not like these bands, don't know if they're your type of music)

Black Sabbath - Ironman
Metallica (best song/s) - Master of Puppets
Megadeth - Gears of War/ Hanger 18
Iron Maiden - Hallowed Be Thy Name/ Wasted Years
DOWN - Pyllamid/ Three Suns One Star
Dream Theatre - Under A Glass Moon
KoRn - Right Now/ Freak On A Leash/ Twist
Killswitch Engage - The End Of Heartache
Machine Head - Aesthetic Of Hate/ Hallowed Be Thy Name (Iron Maiden Cover)
{^^ Don't think you'll like them much, they're quite heavy}

Opeth - Coil/ Porcelain Heart (This band has a mix of heavy and soft songs, so if you choose another song, might not be what u expected)

Ozzy Osbourne - Hellraiser/ Civilize The Universe (Another one who has a mix of soft & heavy)

Prmius - N.I.B (Black Sabbath Cover)




Hope this helps you

you seem to have similar taste to me (just a little lol) so here you may like:

Linkin Park: In the end, Leave out all the rest
Hellogoodbye: Here in your arms
Boy sets fire: Requiem (I love this one ^^
Jordin Sparks: Tatto
The last goodnight: Pictures of you

You can try Nickelback too they have many nice songs. One Republic is nice too or Savage garden ^_^

The rest I hear too is more RNB so I don't think you will like it lol

Well I would support Mia's choice about One Republic and Linkin Park and with Nickelback as well although One Republic and Nickelback are slightly different from Fall Out Boys and Green Day.
Anyway I would suggest something more like AFI December Underground or SUM 41 whichever album you choose, they are something more alike Green Day and Fall Out Boys.
But I would recomend you to visit last.fm and search for band you like on your own for you can listen to parts of some songs and well you can see who are similar artists so I would recommend that above all.

Parts of songs? I go on there and I can listen to whole albums O.o

Well you can listen to whole songs and albums if you have an account but I'm saying if you don't cause as I see lately they had some problems with creating new accounts or so I've been told.