Is there a way to increase the amount of items that I can carry with me in Diablo 3, much like you could purchase more/bigger bags in WoW?
No, like its predecessors you can not increase your carrying capacity. However items take significantly less space in Diablo 3 over Diablo 2 (however more space than WoW), allowing you to carry a surprisingly large amount of items.
You can store items in your Stash however, which can be upgraded in size.
No you can't
There is no way you can increase your inventory size in Diablo 3 (But you can buy extra space for your stash). In Diablo 2 you could use some tricks and Wow you could buy larger bags.
All item take far less space in your inventory in comparison with Diablo2. Now your items take 1-2 spaces.
Independent if it's a dagger or a flamberge will take only 2 spaces. While Belts and potions will take 1 space.
It is not possible to increase the amount of space you have in the inventory of your character.
In town there is additional space for storage in a stash, and as of patch 2.1, there are now a maximum of 5 stash tabs.
If your stash is full, you can always create a level 1 character, have it run to the stash, and pull as much from the stash as possible into inventory as storage.
While this may encourage "hoarding" it is useful if you need more space in your stash and it has spilled over onto your main character's inventory.