Bag making requires runes, but the low level ones are cheap. A 20 slot bag is 10g, but an 18 slot bag is only 2g.
At any rate, the bag sizes are 10,12,15,18,20, according to the 5 rune types.
Different types of bags can be made, which will have different functionality.
10xResource+3xDust+Rune = Safe bag. These are invisible to merchants.
10xResource+3xScales+Rune = Equipment bag. Equip-able stuff you pick up will go here. Armorsmith only.
10xResource+3xBones+Rune (and Tailor) = Craftsmans bag. Ore, Wood, Crafting materials and the like will go here first.
10xResource+3xBones+Rune (and Leatherworker) = oiled bag. Junk trophies go here first.
Regular bags I believe are just Resource + Rune. The others I've listed above have to be discovered in the Discovery panel.
I suspect you're holding onto a lot of stuff you don't need. Pro-tip: take everything out of your bank, and go talk to a merchant. Press the Sell Junk button. If it disappears, you didn't need it. Most armor should be sold, not salvaged, as you'll get more by just buying the material off the TP. Things you need to hold on to are anything that has it's own bank slot, and trophies that are Gold or better. You'll know them when you see them. They'll also sell for a lot on the TP.
Golden Rule: if its not worth anything on the TP, sell it to a merchant. Also, you'll see a lot of people listing items for 1 copper more than its merchant price on the trading post. These people are dumb, never do this: there is a 5% listing fee, and a 10% sale fee, which get deducted from the seller, and that 5% listing fee is non-refundable.