1. Reel to backing.
Arbor knot or similar, same as a conventional. Braided Dacron is the classic backing and will be fine in freshwater. For Dacron, 20 lb below 8 weight is typical. 30 lb for 8 and above.
For gelspun, no lighter than 30. In saltwater, no heavier than 65.
2. Backing to fly line.
The important thing is to make sure that the transition from fly line to backing is smooth so that the thick end of the fly line will not catch in the guides. Ideally the knot will be strong enough that the fly line breaks before the knot.
So Now you have some choices.
I like to make it easy to change fly lines. So I go loop to loop. I tie a Bimini loop in the backing, and the double up the loop with a double surgeons knot. The double loop makes it easier to undo the loop after it digs into the fly line coating on a big fish. The loop in the backing should be longe enough that it can loop around the reel or a coiled fly line.
Fly lines often come with welded loops on both ends. First generation welded loops were unreliable. The new ones are pretty good. I wouldn't trust some guy at a fly shop counter to do an on-site weld. I also not believe in home weld techniques using heat or solvents.
If the fly line does not come with a back end loop, you can simply double over the fly line, tie a couple nail knots and whip the end with some thread coated with a soft adhesive, like something you would use for wader repair (e.g. Pliobond, Aquaseal).
Or even better, make a double catch loop in some hollow braided mono (30lb for under 8wt, 50 for 8 and over). slide about 6 inches up the back ( can get by with 3 or 4 inches on trout gear), nail knot (only one) with 6 lb mono, and coat with adhesive as above. Folks are starting to use hollow gelspun, but I have not tried that yet. Gelspun will not have as much of a finger trap grip on the fly line compared to mono.
Looping the fly line is more important than looping the backing. If you don't change line a lot and are not going for big fish, you can simply tie the backing to the fly line loop with your knot of choice (clinch, San Diego, etc.).
3. Fly line to leader.
I still do a loop to loop. Doubled over fly line, or even better braided loop. With a loop, you can replace the entire leader setup as needed. Keep the loop as small as possible, no bigger than 1/4inch ID. On the fly line side, no more than about 5/8 on the leader.
Loops in the leader end of the fly line horrify trout and some saltwater flats purists. They believe that the loop screws up the layout of the cast (don't think so), and makes more of a splash on landing and can change the drift angle (mebbe true, but pretty trivial).
If you want to avoid a lecture from a local trout guru, nail knot the thick butt section of the leader to the fly line (no loop). The butt should be at least a foot long, so that you can tie on fresh leaders several times without replacing the butt. For some reason, some trout gurus approve a tying a loop in the mono butt

Use a nonslip loop here.
4. Leader construction.
For knots - nonslip loop at the butt end. Section to section of similar diameter, use a blood knot. It takes me some experimentation to find a the right knot for a specific combination, but a slim beauty and RP are often used by me.
In terms of leader section length/diameter formula, Too much to get into here, but it can be quite simple. For 90% of my fishing, I just use a straight shot of mono of the final tippet strength. For tapered leaders, I am of the belief that less is more. Three sections are usually plenty, with the first section being about half the overall length. If you get into one of those ridiculous situations where you supposedly need a super long leader, just add more tippet.
For freshwater stream fishing, especially starting out, it can be easier just to buy a a couple tapered leaders and a spool or two of tippet material. Just tie on more tippet as you use up the tip tying on new flies, or if you need to change tippet size or length.
5. Tippet to fly.
For any fly that is supposed to have an action, there has to be a loop. I use a nonslip loop. For dry flies (not poppers) that are supposed to layout straight with the tippet, use a specialty knot so that the fly is tied straight and not skewed (there are better knots than a clinch for this, but not my knowledge area).
6. Alternate approaches.
There are some more complicated methods that involve stripping off coating, jamming leader or backing inside the coating, DIY solvent or heat welds, etc. These can provide a smooth transition, and may work well for the folks that use them, but you will generally be giving up some strength and durability, which could end up biting you in the a**, or not, depending on the type of fishing that you are doing. So not a good match for the type of fishing that I do.
Always more than one "right way" to do things. I am curious to see what the other fly folks here do.
Clear as mud?
-J