Mounts are in the Collections interface, just click the horse icon at the bottom of the screen, then go to the Mounts tab. I always forget the current hotkey, Shift and I want to say i
For professions, you’re limited to learning two at a time, and you can find trainers in every capital as well as scattered around smaller quest hubs. Right-click the magnifier icon on your minimap to track their locations. Blacksmithing and Engineering are done mostly at an anvil, Mining at a forge, the rest you can do nearly anywhere but they sometimes require special tools. Go to the Professions tab in your Spellbook, click the profession’s icon to open the menu (you can put the icon on a toolbar for faster access, Esc > Interface > Action Bars if you haven’t activated more toolbars yet), then for crafting you can see what’s needed for each recipe, as well as make the recipes if you have the correct items. Generally you’ll want to have matching gathering and crafting professions: Mining with Blacksmithing, Jewelcrafting, and Engineering, Herbalism with Alchemy and Inscription, Skinning with Leatherworking, then Tailoring and Enchanting (make items with cloth that you find, then disenchant it). There’s also Cooking and Fishing as secondary skills, and I remember there being something about Archaeology no longer available to Starter accounts, but you can check that sometime to see.
If you’ll be playing the Starter Edition for a while, might as well go through all of them up to the 100 skill cap to see how each one works. You’ll be able to make a few moderately useful things as well like enchants or armor kits, although with all the scaling I don’t think those have much affect anymore. Potions could still be useful though if you happen to find some good recipes like Swiftness, just keep in mind that when you unlearn a profession to switch to another one, you’ll also unlearn all recipes.
You’re right that the auction house can’t be used without an active subscription, most you can do is browse.
For bags, you can buy one at the Darkmoon Faire this week for a prize ticket, the others you can buy from the faction quartermasters once you’ve gotten their rep to Revered. You can do any available quests in their starting areas, but the quickest is running dungeons while wearing their tabard, and I recommend starting with Orgrimmar (Stormwind on Alliance characters). You’ll be limited to 16-slots for the time being, but it’ll be enough to fill your character and bank inventories.
You can buy a few heirlooms with prize tickets at the Darkmoon Faire, along with other account-wide things like pets and mounts. Fish up 100 Darkmoon Daggermaw (floating debris are good for them, just don’t auto-loot the bloated sharks because sometimes they contain more than 10g), and use them to buy a special treasure map quest item from one of the vendors on the pier, and it’ll lead to 100 prize tickets.
Since you’ll be hitting the 10 gold cap a lot, you can buy some Elemental Flux or Etherium Thread from profession vendors to drop the balance down so you can earn more gold. They stack to 200, and when you have an active sub again you can sell the stack(s) back to a vendor to recover part of that gold you would have otherwise lost. Also, when you’re fishing around Darkmoon Island, don’t auto-loot any bloated sharks because sometimes they contain more than 10g, just check to see what’s inside (there’s an Interface option you can check to adjust how auto-looting works when holding Shift), and if it’s more than 10g you can decide if you want to stash the shark for later.