If you want to be able to lock and unlock it from either side, a mortice lock will be suitable. Unless you are good with a chisel, try to find a chippy to fit it. The stile of the door (the vertical pieces of wood at each side) needs to be about 44mm thick (or more). This is the same as an ordinary front door.
I am very fond of the older Chubb mortice locks. The levers and bolt are brass, and the springs are bronze, so the working parts are rustproof. The case of the older models is thick steel so lasts well. You can usually get one on ebay for £10 or less including several keys. The case of these is painted light grey the model numbers of these begin "3G..." The reason there are so many being sold second-hand is that they easily last 50 years, and people have them spare when they change their front door. Cheaper locks will not last outdoors.
If it is too thin for a mortice lock, you can use a Rimlock which is screwed to the face of the door (on the inside). Most of these are sold for sheds and are flimsy rubbish. However you can get a FB Rimlock (which is a grade used for doors that the Fire Brigade can open with a master key) and these are quite sturdy. Locksmiths and ebay have them.
Some good rimlocks, like the Marston and Wellington, can be opened only from the outside (these are used on Utility Cabins, e.g. electricity substations and gas cabins) so would not suit you.
You can also fit a Yale-like rimlatch. However this will lock automatically when the door swings shut, and can be opened (e.g. after climbing over the gate, or with a piece of string or plastic) without a key. They are also fairly easy to force open by barging or kicking the door.
If you have small children, you can put a bolt on the inside at the top, where an adult can reach it by putting their hand over the gate. This is obviously not secure.