As an artist-
I use D-rings. Single-screw for small pictures, larger two-screw type for large ones. Fit one each side to the frame, just over half way up. B careful to put the screw through the thickest part of the frame, and use a bradawl to make the hole first, so as not to split the frame- a very fine drill if the frame is very thin. Using either wire (old-fashioned), or non-stretch synthetic picture cord (never string as this will age and snap) of a suitable thickness tied between them- through both rings, then tied to itself, tie it hard twice, then pull the cord until the knot is clear of the centre, then tape the ends along the cord.
A picture framer could do this for you.
To fix to wall, if pic is small and light, a single-pin brass picture hook, if heavier, a double-pin hook, if very heavy like, say, an oil painting a metre wide in a heavy frame, I use screw in the wall (drill, fit rawplug suitable for type of wall, then put in a screw ( a no 8 should be enough) with a 'penny washer' on it but leave it sticking out the thickness of your cord/wire, then hang cord over screw and behind penny washer so it can't slip off).
One thing I wouldn't do is perch a picture on a nail using one of those zig-zag things.