A Blacksmith Forge
Smithing is how you make and improve gear. It mostly just costs gold, but it's how you get Ultimate Gear.
Use a Smelter to smelt ore into ingots.
Use a Tanning Rack to tan animal hides into Leather or Leather Strips.
Use a Blacksmith Forge to craft items.
Use a Grindstone to improve weapons.
Use a Workbench to improve armor.
Alvor the blacksmith
Talk to Alvor the blacksmith in Riverwood for a crash course in Smithing.
Use a Blacksmith Forge, Grindstone, Smelter, Tanning Rack, or Workbench for the first time.
Check Blacksmith Forges, Grindstones, Smelters, Tanning Racks, or Workbenches under Help in the pause menu.
Workbench
A Grindstone or a Workbench.
The item to be improved.
1 ingot (or piece) of the main material of the item to be improved.
A Smithing skill of 100, with the perks for the material you're working on and Arcane Blacksmith.
A full set of apparel of Fortify Smithing.
A Potion of Fortify Smithing. If you work quickly, one potion should be enough.
Equip the apparel of Fortify Smithing.
Read The Sallow Regent and choose Seeker of Might (if you haven't already.)
Drink the Potion of Fortify Smithing.
Improve the items!
(Don't forget to switch back to your armor.)
Smithing
The rate at which you learn Smithing depends on the value of the items crafted.
Optionally, activate the Warrior Stone for 20% faster learning.
Optionally, chop some Firewood for arrows (if you have the Dawnguard DLC.)
Optionally, sleep in an owned or rented bed for 10% faster learning.
Smelt your ore.
Tan your hides. Cut 1/3 of your Leather into Leather Strips.
Start crafting:
Jewelry
Arrows or Iron Daggers
Leather Bracers
(Don't forget to switch back to whichever Guardian Stone helps with Archery.)
Training Smithing to 100 is a slow process. If you have a good economy going with Alchemy, you can pay for training (seek out Eorlund Gray-Mane at the Skyforge in Whiterun) and buy expensive ingots and turn those into bows or arrows. You can improve the bows at a Grindstone for even faster learning.
Whiterun
Smithing is really slow to improve. You can skip a huge chunk of grinding by paying someone to train you if you meet these prerequisites:
You need the Merchant perk. This perk becomes available once your Speech skill hits 50. Just keep buying and selling until you get there.
You must be a member of the Companions. Go to Jorrvaskr (the enormous beer hall) in Whiterun and talk to the people there to get started.
Then...
Talk to Eorlund Gray-Mane at the Skyforge in Whiterun. He is a Master Smithing trainer, meaning he can train you up to 90. (He only trains Companions.)
Buy five levels of training from him.
Level up (so you qualify for more training.)
Sell him potions to get your gold back.
Repeat.
Gold Ring (2)
Craft the most expensive stuff you can. If you don't have any gems, craft plain Gold Rings. You get two rings per ingot.
Iron Arrow (24)
You need arrows anyway, and they're a decent way to learn Smithing.
Unfortunately, you'll need lots of Firewood. Chop wood at a Wood Chopping Block (or order a follower do so--they'll get their own axe.) You may wish to use a console command to hurry this along:
player.additem 6f993 100
Use the Firewood and your leftover ingots to forge arrows.
If you don't want to bother with Firewood (or don't have the Dragonborn DLC,) forge daggers.
A Tanning Rack
Loot and buy all the Leather, Leather Strips, and animal hides you can.
Use a Tanning Rack to turn your hides into Leather. Turn about a third of your Leather into Leather Strips.
Craft Leather Bracers.
Enchant the bracers before you sell them. They're not worth much otherwise.