It's a good idea to let a toddler have his/her own bag. For my boy, he's happy to carry his own bag - a children backpack. By carrying a bag, he is trained to take a responsibility to look after the bag on his own (and the contents inside it). Not to weight too much on his shoulder, normally I put one book, one piece diaper, one water bottle and one piece of handkerchief. I will tell him to carry his little backpack when we are going for occasions or shopping. Sometimes he will ask for his backpack if he sees me holding a nice bag or sees us dressing so formal.
I just introduced this french brand - DPAM (Du Pareil Au Meme) in my previous post. I bought a DPAM's Rhino backpack for my boy last year. I like the design and I think my boy will like it. Yes, he is. See how co-operative he was when I told him that I want to take a picture of his little Rhino's bag.