1. A good leash- While I appreciate Dollar Tree for inexpensive items for the dogs, Their leashes are at best temporary as well as their collars. I personally use a Fozzy Dog leash, as it is sturdy and has a convenient pouch built in to carry poop bags, treats. There are other brands out there just as good, FozzyDog is my preference. And the have a great double leash That allows two dogs to walk comfortably, as well as mini leashes for the little guys. A bit expensive, but well worth it as it seems to last forever. But look for a leash that is sturdy, has good snaps and not too long, so you can control your dog- 48" seems to be about the right length. And please stay away from thoseretractable leashes, especially if your dog is apt to bolt suddenly.
2- A good Collar or Harness- This does not need to be expensive. Just look for collars that , if they are adorned, like with jewels or studs, don't have protrusions on the inside of the collar, (it can rub and irritate the pooch) I usually buy mine at the Burien Farmers Market, there is a lady that sells collars there. But most pet stores should have them in stock, and should be able to help you with sizing.
3. Stainless steel water bottle Face it. not every destination is going to have water for your pooch. I find that having a Stainless Steel water bottle is a good answer. There are a lot of plastic water bottles for dogs, and I like a lot of them, but find the leak or get destroyed.
4. Collapsible Dog Bowl- A great thing to put your water for Fido in! Cheap, easy to fit into a pant pocket or on a leash sometimes I take two- one for water, one for food