What schools in your opinion should an infantryman have or strive to have?

The career map you can access on ACT! Look at the attached image.

Ranger, Air Assault, Pathfinder, Mountain Warfare, and whatever they’re calling Small Arms Master Gunner these days would be a great start.

As an 11C you should try for an Airborne Assignment as a junior enlisted. Once you become a SSG you might not get the chance.
Schools you should go for at first would be EIB, Ranger, Combatives level1 and Air Assault at the SL1.
SL2 you should be looking at IMLC, Pathfinder, Jumpmaster, Rappel Master, Combatives level 2, any marksmanship courses, EMT - Basic, and possibly RSLC.
SL3 you want to develop your institutional knowledge base with schools like Battle Staff, Master Fitness, Master Gunner and Master Combatives

Those like SGT Phair said plus add some others that build outside of 11B stuff, like commo any radio classes/schools are huge. So many guys have no idea how radios and other equipment works and it's a huge asset. My point, don't just be another "hardcore" infantryman, become an asset to your organization

Earn your EIB as soon as you can, take on positions of greater responsibility as soon as you are ready, develop your leadership skills. Go through all the manuals and regulations and learn a bit at a time, do your best to be a SME (Subject Matter Expert) There are so many toxic leaders out there you should carve out your own bubble to make your fellow troops motivated even as a PFC. I was a team leader as a PFC for a short time then as a SP4. Develop your leadership style by taking from good leaders around you and adding your unique flavor to it. Do all of this under the watchful eyes of your COC. and you will get that slot to BLC etc. Once you have done this all those good to go schools will be a closer possibility and you will have proven your metal. Above all don't be afraid to fail as long as you learn from failure and do not repeat.

Any school that is offered to you! Look up the glide path for cmf 11 and it will tell you what things you should be accomplishing or have accomplished at each rank. Go to Ranger school! Airborne/jumpmaster, air assault/rappel master, pathfinder, rslc etc...... if you can get it, just go. There’s a lot of slugs who don’t want to do anything but show up in the right uniform at the right time. Like the other guy told you. Be an asset to you unit!

Your unit will have school prioritization for what they want/can send you to. Focus on those first, as the unit will be willing to send you off to the ones they can afford and want you to have. For example, one of the units I was in pushed to send soldiers to Air Assault and Mountain Warfare. Much farther down the list was Airborne, Ranger etc.

As a Veteran of the D.M.Z. I give the following suggestions. Become the best rifle shot you can be. Then be ready mentally to use it. All the world will come alive and seem like it is against you. It is not the world. Only the enemy. Artillery can and will settle things. Just make sure they are on your side. Grids mean lives. Take every tactical class from Veterans you can afford. Time or money. It will save your life someday.

Ranger School will give you the most training to enhance the skills you already have. Master Marksman Trainer is a great school but slots are limited.

Airborne, Air-assault, Ranger, Combat life-saver. SF Selection.