A private teacher would be a great help if you have talented music teachers available in your area, and it really helps if the teacher actually has experience in the style of music you want to learn. Too often, this is not the case. There are resources available online, but any goofball can post on the web so one must search for a teacher online with direction and discernment.
The easiest way to learn any style of music is to start with listening. A sax player wanting to play rock or pop sax needs to listen to that music, search out these styles, and listen to music with modern players that excel in the style. Today with the Internet it is so much easier than it was years ago!
Since the saxophone is not a regular part of every band, you will also want to listen to the other instruments that play similar roles in the musical style you're exploring. Guitar is a favorite instrument for many saxophone players to emulate, since the guitar can play with so much passion and aggression in modern music.
Listening to music is highly underrated as a way to grow your skills. I think that you can improve with just listening (for a while), and you can practice every day and never really get any better. Practice makes permanent. If you do something wrong 1,000 times, you will be even more likely to do it the exact same, wrong way again.
We listen to learn a style of music, then practice to emulate it. Once you have an idea of what the saxophone sounds like in rock music, you then need to try and make your sax sound like that. Have you heard a great lick in a solo? Learn it. You can figure out the notes. Then try to get the articulation and feel to be the same as the example you've been listening to.
We learned to walk and talk by watching and imitation; the principle in learning to perform music is the same. All you need to do is find a model in music that you like and work at playing the same on your instrument. If you continue to listen and let that style of music get into your head, it will start to come out through your fingers over time.
The next step might be looking at your setup and equipment issues to dial in a tone and sound for rock and pop styles. An old, fluffy jazz tone will need to be tightened and brightened up.
Reeds are the first place for sax players to start. Plastic Covered Rico reeds or other synthetic reeds will likely give you a more aggressive tone that will cut thru in louder bands. A higher baffle on a metal mouthpiece can increase volume and projection too. If you are listening, you will have a sound concept developing that will suggest a few changes along the way.
Many sax players will eventually look into electronic options for the sax. Effects pedals like the TC Helicon VoiceTone Create XT or Digitech Vocal 300 can add all kinds of cool sounds to your sax. Simple reverb and delay will add a ton of color and vibe. Once you have some effects to play with, you can begin to color and change sounds like a guitar player does, from clean to dirty, and then big for the solo.
Any effects might imply a need for a little system for you to be able to hear yourself better and a personal monitor. My sax rack has a little mixer, TC effects, a tuner, and a small power amp that feed a speaker only for me. A simple foot pedal and powered monitor can make a huge difference.
In electronic world, there are also instruments made for sax players. The Akai EWI Professional Electronic Wind Controller and the Yamaha WX5 MIDI Wind Controller give saxophone players an electronic instrument option with a much quicker learning curve than trying to learn to play guitar. An electronic wind controller is not a replacement for a saxophone, but it can add to the color arsenal a sax player might bring to pop or rock settings.
The last step in venturing into new musical territory is networking. You need to find people that play the kind of music you want to play in your area—online and in clubs or rehearsal spaces. As you meet folks, let them know you want to jam and give them your contact info. Diligence will pay off!