Zine Making/Selling or giving copies to people
Most people who make zines aren't doing it as a profit-making enterprise, and if you are hoping to make a living from your zine then you're probably wrong. Most people who sell their zines put them at a price which kind of approximately covers the costs of making/distributing them, so that the zine-making process doesn't bankrupt them. A lot of smaller zines (e.g. ones made from a single page of A4) are given away. It's much less hassle not having to collect money etc. And a lot of zines are swapped for other zines rather than bought. It's up to you what you do, but don't expect to see some kind of profit margin on your zine; that's usually not how it works.
Record shops / book shops
Lots of independent record shops / book shops are happy to carry zines. Some of them have stands for them and have a sale-or-return procedure in place to let you sell them. But even if they don't, you can often ask your friendly independent record/book-shop person if they wouldn't mind having them on the counter or wherever.
Sale-or-return is the typical agreement that shops do: they'll collect the money on your behalf, and you can come back in a month (or whatever) and collect your money (hopefully) or your unsold zines. Some shops have a time-limit, for example if you don't collect them within 6 months they assume you've forgotten about them and get rid of them. Some shops take a cut of the profit, and some don't.
People who like zines and who really enjoy photocopying often set themselves up to distribute other people's zines - these are often called "distro"s. There are some large and well-established distros and millions of tiny one-person distros, but typically it's just a case of if they like it, they'll be happy to carry it for you. If you find distros which have stuff you like, or your stuff fits in with them, just ask...
Remember that it's typically all done for the love of it so it's not quite the same as approaching some publishing company. You don't need to be particularly formal, just send them a zine and/or have a conversation with them.
People who like zines and who really enjoy herding cats often set up a "zine fair" or "zine symposium" or some such, in their area or for their particular scene. Have a look around to see if there's anything suitable.
Carrying zines around
If you've got a nice load of zines freshly made and you want to take them to gigs, fairs, etc, you probably want to keep them looking nice, so it's a good idea to find a sturdy box to carry them around in. If you've only got a few then anything will do, but if you get to the point of having a lot to carry around then finding an old suitcase, camera-case, or some such case with a handle or strap can be really helpful.