The bare essential equipment for taking your dog for a walk is a leash and something to attach it to- but it can get much more complicated pretty fast. For most dogs and their humans a harness is more comfortable and offers better control than a collar for walks. A collar is mush more likely to slip off; in flat faced breeds it can exacerbate breathing issues; and in other breeds can cause injury to the trachea and neck when they pull.
So harness it is- look for a harness that is washable, that fits securely without edges pressing in to their sides or under legs. A leash attachment on the front is a must- it is much easier to control and teach your dog not to pull leading from the front.
I have used a few different harnesses, I like some for some dogs and not others-depending on their sizing and the dogs behavior. Max wears a very minimal web harness I think it breaks my rule of having a front leash attachment, because max walks very nicely loose lead and mostly always has. LeeLoo has a great harness from Rough Wear- she is on her second one due to size- it is padded, very sturdy, leash attachments front and back and it's orange so I can see her better in the woods when she is off lead.
Previously I had a dog that just would not stop pulling no matter what, I used a specialty harness from Sporn- that puts pressure between the shoulder blades when pulled. The first time I used it she was like "oh you mean i'm on a leash attached to you - oh ok" it was very dramatic and solved our pulling issue.
Leashes are not so complicated and can be more personal preference. You must have a standard 6 foot leash for training that length just works really well. There are lots of patterns and materials out there. The one thing to be mindful of is the weight of the leash; your tiny puppy needs a lightweight thinner leash than your 6 month old golden or your 18 month old hound X. I am a bit of a leash collector... but what I mostly use is a plain webbing leash with a plain snap clip. I do use biothane long field lines but I'll cover that separately.