Disturbed, Beatles, Sublime(r.i.p. Bradley Nowell, and the goo goo dolls.
Early or later Beatles? I don’t really listen to those other bands, so I can’t make recommendations from them…
If you like Sublime, how about:
Save Ferris
No Doubt
Archers of Loaf
If you like Disturbed, listen to System of a Down. They take a while to get into.
Specifically, the song Chop Suey. It blows my mind every time.
Los Fabulosos Cadillacs - Los Fabulosos Calaveras would be the best album
All roads lead to Frank Zappa.
I like both.
Offspring?
here are some i always recommend:
The Black Crowes
The Jayhawks
Ben Harper
Wilco
Jeff Buckley
You might like the newer material from Anything Box.
If you like Disturbed, you might check out Into Eternity, one of my favorite bands. Their only album to date is a bit weak lyrically, but the musicianship is pretty amazing.
Dang, Splanky, 311 is a perfect recommendation! I’m surprised that one didn’t come to mind earlier.
Although they don’t really sound like any of the bands you listed, I just wanted to remind everyone that The Pogues are the greatest band ever. Buy all of their stuff. Listening is optional.
according to yoru likes… it seems you could flick on a top 40 station and be pleased.
just for fun I’ll mention…
Gwar, Fantomas, Rufus Wainright, Elliot Smith, Blur, Oasis, MC Solar, Faith No More, Tool…
I don’t know… your likes are kinda vague
According to Audiogalaxy Rhapsody (the coolest music program–ever):
Disturbed is labeled as “Rapcore”.
Contemporaries include: Coal Chamber, Limp Bizkit, and Korn
Influences include: Anthrax
Most popular “Rapcore” artists according to Rhapsody usage (not including previously listed): Kid Rock, Linkin Park, Incubus, Rage Against the Machine, 311, Saliva, P.O.D., Insane Clown Posse
Sublime is labeled “Ska Punk”
Contemporaries include: Goldfinger, Less Than Jake, and No Doubt
Influences include: King Tubby
Most popular “Ska Punk” artists: The Clash, Bowling for Soup, Rancid, Reel Big Fish, Save Ferris, and Fishbone
Goo Goo Dolls are labeled “Adult Alternative”.
Contemporaries include: Sheryl Crow, Soul Asylum, Semisonic, and The Wallflowers
Followers: Uncle Kracker
Influences: Buzzcocks, Cheap Trick, The Replacements
Most popular “Adult Alternative” artists: Coldplay, Norah Jones, U2, Tori Amos, Sting, John Mayer, Sarah McLachlan, and Gretchen Lieberum
And, just because I recommend them to anyone with ears: Modest Mouse and Built to Spill.
Hope that gets you started 
For Sublime connections, I would second both Fishbone and Less than Jake, and also mention Slapstick.
Fishbone, like Sublime, combined many different styles of music (including ska, punk, soul and funk) very well, their best album being “Truth and Soul”.
Less than Jake started out as generic but great skapunk and have evolved into ska/punk/rock/metal/etc., catchy, anthemic songs with great vocal harmonies despite not having virtuosos at vocals; their best album is “Borders and Boundaries”. I first got into them with their collection of early records compiled as a CD, and they have gotten better and better.
Slapstick only put out one album, but it was the epitome of skapunk. Great, fast ska songs, with solid horns and thick guitars featuring plenty of pick slides (punk rock!!!). Most of their songs can be found on their self titled discography.
The Clash were already mentioned, and if you don’t have any of their stuff, you really should look into it, as Sublime and 311, like many bands, were heavily influenced by them. Other important bands that I assume you know about but will mention just in case are Operation Ivy and The Specials, the two major influences (aside from the multi-genre Clash) to skapunk.
You should check out Jack Johnson. He kind of sounds like the slower Sublime songs.