I've taught some classes for a mountaineering club, and the following sequence is based loosely on what we do.
(1) Learn traditional map-reading and navigational skills. Move beyond attempting to navigate the way most people do these days, by peering at the screen of a cell phone. Get in the habit of printing out a paper topo map for any new route you're doing, and build competence at relating the topo map to the landscape you see around you, without depending on GPS (which is limiting and unreliable). Build experience with cross-country travel, so that you're not freaked out by going off trail or losing a trail.
(2) Learn very basic skills at avalanche avoidance. The most basic skill is extremely simple and does not require taking an avalanche course or buying gear. a) Has there been 15 cm (6") of snow in the last 48 hours? b) Is the slope 30-35 degrees or more (or is the area bare of old-growth trees)? If both of these factors exist, don't go.
(3) Build experience with the ice ax, and also with crampons, microspikes, and snowshoes. The ice ax is a fundamental tool of the mountaineer. Practice self-arrest. Get comfortable with snow travel techniques like the plunge step. Learn to move confidently on steep snow, using the appropriate techniques. While you're gaining this experience, you'll gain skills in keeping yourself comfortable and happy in a cold environment.
(4) Learn basic roped climbing techniques. You should know how to put on your harness, half a dozen of the most important knots and their uses, how to flake the rope, and how to provide a hip belay or a belay using a belay device. This could be in the context of rock climbing or snow climbing.
(5) Learn to lead climb (on rock, snow, or ice), and to build anchors.
For instance, as a beginner - is learning to be a strong rock climber a good skill to master to eventually learn ice climbing. Or as a beginner is it better to take glacier routes (and learn glacier travel/crevasse rescue) and over time learn to master more technical climbing.
You can do rock, ice, and glacier travel in any order, or learn any subset of these skills that you wish. They overlap a lot, because they all involve a lot of the same roped climbing skills and anchor building skills.
First aid training is great, but this is really more of a general life skill. Even if you take a wilderness first aid course, you're more likely to use the skills when you're in your home or at work.