I am a lvl 10 Arms Warrior. I was wondering if it is possible to buy equipment and weapons from a shop? Or is the only way to get new items by doing quests and getting drops from monsters?
Yes. You can buy items from shop. While in Stormwind you can press 'm' to show map and then select to show you vendors. If you are looking for Heavy Armor, which I think you do, you can check the Stormwind Heavy Armor Vendor.
However, at your level it's not likely to find better armor in shops than what you get from completing quests(until later when you will have faction vendors). Also, you might check out the Stormwind auction house where you might find some good stuff.
From level 15, some of the best gear you can get, will be looted from dungeons (try RDF).
Other than this, most likely you will find Blacksmiting a very good profession for your Warrior. Professions help you a lot to gear up to lvl 85. Hope this helps in a way. If you have any other questions feel free to ask.
PS: You seem new to WOW so maybe try some of the guides out there, they really help a lot.
One thing to consider when looking to buy gear is how long you will be wearing that gear. At level 10 (and really sub 50), you'll be leveling pretty quickly, so you'll naturally come across better gear due to your increased level, higher level quests and instances.
Honestly, at level 10, I'd do another 5 levels or so of quests and then start hitting up the Dungeon Finder. Instance drops are usually significantly better than most quest rewards or random drops and usually better than stuff you can buy from vendors or on the auction house.
The only items that I typically look to buy at low levels are the odd slots - neck, rings, trinkets, since there aren't many drops of those in 1-60 content.
Realistically speaking, to generalize points somewhat highlighted by the other answers, there is absolutly no point to buying any generic (White Name Color) item from any vendor, ever.
The items sold by such vendors are both ridiculously overpriced and sup-par to any item you will get either from a quest reward or from a dungeon drop.
I've personally played WoW since before it was released (I played the open beta) and the only time I've seen it as a good idea to buy white vendor items is when I make a new character, and can give said character large amounts of gold from my other characters. At that point money is not an issue and buying the character a full set of gear right off the bat speeds up my early leveling a bit until I reach the minimum level to start whatever dungeon I choose to start.
Brandon does have a good point in that sometimes you can find items from vendors that are otherwise rare as drops until higher levels, ie: neck peices , trinkets ect. Those would be pretty much the only exception to the rule.
Auction House purchases are also exceptions to this rule, as they are not 'vendors', but rather a hyped up trading system between players.
Keep in mind as brandon said, that you will be leveling very quickly, so quickly in fact that spending a large amount of money on ANYTHING before your at least level 60 or so is a waste of time, as you will be replacing your gear on a rather constant basis, so everything is pretty much a 'hold-me-off-until' type of item.