Weeeeeeell, there's really no way to play Railjack solo, so you're going to want to switch to public, then go to the console in the centerish of the bridge, which is not the hole in the back of the bridge or the control harness at the front of the bridge, to bring up a navigation menu and pick a mission in Earth Proxima, preferably one of the lowest level ones. And then you play the same damn mission over and over again, a space Exterminate with a warframe Sabotage segment inside an enemy asteroid base or ship, getting resources and fixable parts and space mods and space endo to upgrade your railjack and make it not die quite so often. In the process you'll also gain Intrinsic Skill points based on your affinity (XP) gained, which can be spent to progress along one of four skill progressions. I'd go for Engineering 1, Gunnery 2, Tactical 4 as a first set of goals, because those are some of the most necessary skills on offer for a total of 18 points, but since you're hosting with your own railjack, you'll probably end up flying it and might want the first couple of ranks of Piloting even before those.
Of course, you can also join random public missions from your orbiter just by setting to public and clicking the railjack in the upper right corner, such that someone else will be bringing their ship instead. You'll have the same opportunity to progress in all of the same ways, but you won't have to use your own potato ship, you won't have to fly the ship, and you won't actually need anything except the Intrinsics points since all of the resources in railjack are for upgrading your own railjack.
Within the mission, you'll mostly be either flying the ship, attempting to kill enemy fighters either from one of the gunnery stations (don't bother until you have Rank 2 Gunnery) or from archwing (which is honestly the better bet if you have a decently built up archgun unless the railjack has very, very good weapons, but building archguns for railjack is weird too so look into that), killing crew ships by boarding them generally using archwing and destroying their reactors, fighting off boarding parties on the railjack and putting out fires, patching leaks, etc. with the Omni tool in your gear wheel, or doing some basic Sabotage mission stuff within an enemy installation. If you're host, you'll probably be flying the ship from objective to objective and trying not to die while shooting some things, and since you start out with a slow ship and terrible weapons, it'll feel even worse than it does once you have the best possible ship and weapons, which is still godawful.
If you'd rather play solo, you'll have to wait until DE releases the Command intrinsics progression, at an unspecified future date.