You have several options on how the bag is best carried. Think about what you will be doing (and how active you will be getting), as well as how comfortable you feel each option will be.
A carabiner clip is similar to the one which would be used on hiking/climbing gear or secure keyrings. They are slim yet very strong, ideal for fastening on several optional surfaces such as around belts, onto rucksacks, on belt clips or even onto their lead handle.
A belt clip will be a plastic clip on the bag which won’t allow the bag to dangle, which is a good idea if you will be active such as running and don’t want it getting in your way.
You can buy bags which go around your waist using a secure clip fastening. These can often be adjusted in length, but ensure it can go as short or long as you require.
One which will go around your body or shoulder will be handy if you won’t have a secure waistband, belt or other section to clip it onto, and you want it to be secure to your body without rubbing on your waist.
The dimensions of the bag will, of course, depend on how much you need to put in it. Think about what you need to carry such as bottles, treats or balls/toys.
But also bear in mind how bulky this will make the bag. You may even feel you want to carry a bottle separately to keep the bag as small as possible if you will be more active.
The dimensions of the bag may also change the carrying option you choose. A big bulky bag may feel more secure on a shoulder strap tight to the body compared to on a carabiner clip, for instance.
You may wish to get something which is both waterproof and machine washable if you will be in muddy, wet conditions. Getting a liner which can be cleaned separately from the bag is a great idea as any crumbs or food smells can be disposed of.
Think about what you need to carry, too. If you will be taking a key, phone and a bit of money with you (as any dog walker should), look for a secure pocket large enough to fit these in. This will keep all in one place and should be safer than putting them in any of your pockets.
If you will be walking at night, a reflective bag will give you some added protection when darkness falls.