Definitely start with the fundamental skills that transcend each character. This would include short hops, wave-dashes, L-Cancels, and dash dances. Since ICs utilize their wave dash and have unique properties associated with their dash dance, an argument could be made that these skills are all incredibly important to make use of. If you're just starting, it's important that you familiarize yourself with these techniques and to frequently try to implement them into your game so that they become second nature. These skills will develop overtime naturally, but it's also important that you spend some time practicing them as well, if at all possible.
Once you feel comfortable with your movement, then I'd recommend starting to dabble more into Ice Climber specific technology, such as desyncs and wobbling. Some people warn against learning to wobble early into your ICs career, but I personally think it is important to understand how and why it works. This can help prevent mindlessly trying to wobble off every grab you get, but it doesn't remedy excessive grab attempts, so be wary of that.
Here are some guides to help you start out:
SleepyK's SSBM Competitive Beginner Guide: Incredibly dense, but an awesome resource. Check out the practice section, specifically.
Derf's Ice Climbers FAQ + Useful Links: I very useful, comprehensive list of all the main Ice Climber guides out there to help you get started. There's a lot of material, but all of it is very informative and useful. Also, it's listed in a suggested reading order, which is something that you specifically asked for!
Hope I didn't yak your ear off! Good luck with practice and have fun!
