So I have about 10 iron ingots, but not sure what to start using them for first. Which is the more important thing to invest my ingots? I know armor swords and tools are important, but what is my priority?
Lots, and lots, and lots of things.
But let's give you a priority list, since that's what you're looking for. This list assumes you are playing the "Vanilla" version of Minecraft, with no mods installed.
Which is to say, a pickaxe first. Your Iron pickaxe will be your friend deep underground, when you find a rare ore to mine such as gold or, dare I say, diamond.
BUT, you generally only need one for those rare ores, especially when you have a low number of ingots.
An iron sword is also a major improvement over the stone sword, and if you find yourself troubled by monsters frequently, it could be a worthwhile investment.
Don't waste the iron on axes, hoes, or shovels though. While they will last longer and work a bit faster, these are luxuries rather than necessities. Also, don't throw your stone pickaxe(s) away either. Those are your work horses, and they should still be used to clear out large swathes of weaker ores and stone blocks.
Eventually you will find diamond, and a Diamond Pick will become your go-to pick for mining out rare minerals, and a Diamond Sword your go-to melee weapon. They will last a long, long, LONG time and are very efficient, but can still eventually break. So you may want to keep your Iron Tools around anyway, and may even consider them as your faster workhorse once you get a lot of iron.
After an iron pickaxe, some armor is probably your next most important thing to build. Unfortunately for you, it takes 8 ingots to create iron breastplate, and you just used 3 of your 10 ingots on a pick!
But, any armor is better than none at all, which is why you hopefully already have some leather armor. Replace your helmet, pants, or boots with some iron armor. An efficient way to do this is to make yourself an iron helmet, then save the last two ingots for a sword. But you can also spend your 7 ingots on iron pants, which will offer you more protection.
You probably don't have redstone yet, but once you do, you can use four iron ingots and a single piece of redstone to make yourself a compass! (Or was that four redstone and an iron ingot?) This is handy for long journeys, as it will point you back to your spawn/most recent bed. Though it's not a necessity, since cobblestone can be used as a makeshift compass. Still, a handy tool to have for longer journeys.
You wouldn't think something so innocuous as a bucket would be so high on the list, but it's actually a fairly useful tool. It can save you from an unexpected lava flow (especially if it is full of water), and can be used to move your farm water to a better location. Not a necessity, but definitely nice to have.
Remarkably cheap, and makes your sheep a lot more efficient at resource-dropping. A convenience tool, since you can still get SOME wool from killing a sheep, but good for harvesting bedding material.
While setting things on fire is cool, and is the quickest way to clear out trees, you won't be getting a ton of use out of this item, though it can be used as a handy sustained-damage weapon against mobs. And it has a relatively low cost of just one ingot and one flint.
It would be a bit of a waste, but if you're REALLY inclined, you can use 6 iron ingots to make yourself an iron door. Build yourself a button, lever or pressure plate next to it, and you've got yourself a zombie-proof entryway to your home, and 4 iron ingots left over for that pick you need! Though really, this should wait until you've got more than 10 ingots.
Which covers all the basic, relatively low-cost items that you can make with just a small collection of ingots, in order of importance. When you get more ingots, there are of course additional recipes, but you should probably save up quite a lot more than ten before considering these. Those recipes include:
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Tools, then armor, assuming you're mining. If you're taking a more aggressive surface world approach, then a sword, then armor.
An iron pickaxe can mine gold, redstone, lapis, and most importantly diamonds.
Diamonds are the next tier above iron tools, and last for significantly longer. Once you have a diamond pickaxe, then you can spend the excess iron on armor.
Once you have significant stockpiles, you might want to construct a mining cart system.
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The first thing you want is an iron pickaxe. That way, you can mine for diamond.
After you make a pickaxe, you want a sword, for protecting yourself.
Once you have plenty of iron (which should be easy to get if you go caving, which you should), you should strip mine at y=12 for diamonds. Be sure to get full iron armor and a water bucket first, in order to protect your self from mobs and lava.
10 iron ingots...
You won't really need to return to the surface for days.
iron ore is much more plentiful down there (as is any other kind of ore) than near the surface and when it shows up there are typically more blocks of it together.
Don't work your pickaxes til breaking. Use each until its' health gets red then combine them to make a repaired one.
Before long you will have more iron than you know what to do with.