What is the basic technique for making tea with teabags and a mug?
I've never known the science behind it, but water heated in a microwave oven makes horrible tea and coffee. You need a kettle.

The standard British teabag-and-mug technique (as opposed to the loose-tea-and-teapot technique) is:


put cold tap water in kettle
turn kettle on
put teabag in mug
allow kettle to come to full boil
fill mug with freshly boiled water
leave for 30 seconds or so
remove teabag with a teaspoon; give it a little squeeze for extra flavour
stir in sugar (optional)
add milk (optional)


Scientific rationale:


The water needs to be as hot as possible to extract all the flavour: boiling water can't get any hotter
Remove the teabag before adding milk or sugar because otherwise some of the milk/sugar will be removed along with the teabag
Stir in sugar before milk because it will dissolve more efficiently in hotter liquid
Milk last because you can judge the colour more easily


However

I've seen people claim that the water should be cooler than boiling, because the boiling water destroys subtle flavours in the tea. That may be true, but I suspect that teabag-grade tea just isn't that fine; in any case the conventional wisdom is boiling water for tea, below-boiling water for coffee.

A note on strength and timing

The longer you leave the teabag in, obviously, the stronger the resulting tea. Experiment and find your preference. 30 seconds seems to be about right for a typical British tea drinker, and a typical British teabag.

However, it should be noted that a typical British teabag isn't really intended for the one-mug method -- it's a size originally sized for teapots, and you're likely to get at least two mugfuls from one teabag, if you make tea in a pot and top up with boiling water after pouring.

You could, in theory, re-use a teabag to make a second mug of tea, but teabags are so cheap that hardly anyone bothers.

The teabags found in some cafes, smaller and with a string for pulling them out of cups, need a longer steep, since they contain less tea.