I think the most important things are to communicate in good time and to look out. Listen out as well as transmitting intentions in order to build up a good mental picture of all the other aircraft in the circuit - if not sure ASK! Most of the conflicts/problems in circuit joining come from not doing so. Provided that both are done, one can safely fit into a circuit just about anywhere, subject to any over-riding local procedures of course.
For many years the RAF has taught a "standard join" for a busy circuit, which consists of running at circuit height parallel to the runway in use, just offset to the deadside. This allows flexibilty in turning downwind to fit in with other traffic. Any extension of the pattern occurs upwind, rather than messing up the pattern for others by extending the downwind leg. The procedure starts over "initials" which is a point just on the deadside, about 2 miles out from the threshold. Obviously, one does need to get the correct runway.....
BTW, the "run and break" which people get all bunched up about, is just a modified standard join, using more speed to run in and join and a lot of bank and airbrakes on a closed throttle during a more punchy turn to downwind.