When my glass kettle is boiling water, you can see all the bubbles going up in a line.
Here is a video of my kettle a moment after it turned off.
A glass or metal kettle contains many tiny cracks and pits and crevices in its surface which are too small to see but which contain very tiny amounts of air in them. These air-charged pores act as nucleation sites for the phase change water -> vapor, where the vapor is preferentially generated. As long as the pit retains a little air, it will continue serving as a bubble-generation site; when the air gets used up (a little escapes with each vapor bubble) then the pit is deactivated and the bubbles stop.
The size of the pit mouth, the surface tension of the water, its density, the value of gravity and the ambient temperature of the water itself establish the size of the bubble at the moment of detachment from the wall which means each pit will produce a fairly uniform stream of bubbles.
Larger pits become active at relatively low temperatures and smaller pits at higher temperatures. Since kicking a pit into action requires a bit of time, you can activate the smaller pits first by heating the water fast enough to outrun the kinetics of bubble formation from the large pits.
This effect also pertains to beer in a glass, where the beer is supersaturated with dissolved CO2 and the exsolvation process poccurs preferentially at air-filled pits. If an advertisement photographer tries to take an appealing picture of a freshly-poured glass full of beer and is using a brand-new pilsner glass, no bubble streamers will occur because the new glass contains almost no pits or crevices. To remedy this, steel ball bearings are first shaken inside the dry glass to knock pits into its surface. a glass prepared in this manner will produce generous amounts of bubble streamers when filled and then photographed, which make people like me want to buy and drink that brand of beer.
There are classes of surfactant molecules which are especially good at deactivating nucleation sites by rendering the pits wettable, expelling the trapped air. you can see how this works by boiling pure water in a metal kettle and then trying it with surfactant-spiked water- the difference is dramatic. By the way, those surfactants are persistent, so the experiment must be done with pure water first.
Niels' answer already explains how the bubbles at the bottom form, so let me take a crack at the second part of your question.
It's often taught that it takes an increase in temperature to make water boil. That's not actually true - and while it's far less visually impressive than the image of "boiling water", normal evaporation is also water boiling (turning into vapour). Liquid water spontaneously evaporates, lowering the temperature of the remaining liquid. How fast water evaporates depends mainly on temperature, pressure and relative humidity. The lower the pressure, the lower the boiling point.
When you heat water from the bottom, you create convection currents - warm water raises to the top, and is replaced by colder water. The top is where much of the cooling happens - the warm water evaporates quite rapidly, cooling the top surface. If there's enough water evaporating, you'll observe "clouds" rising above the pot - these actually aren't water vapour or steam, they're evaporated water that has already condensed back into the liquid, but in small droplets that are easily carried upwards on the warm air (as the water condenses, it releases a lot of heat into the air).
The vast majority of the heat loss (and the evaporation) is driven by this liquid convection. The visible bubbles play very little role, and indeed, they don't even have to be there at all - as Niels correctly points out, they form thanks to imperfections in the surfaces (or impurities in the liquid). In fact, water does turn into vapour on the bottom - but there's two very important forces that oppose that. Surface tension and pressure.
To get vapour on the bottom plate, you must overcome the pressure of the surrounding water - steam is far less dense at constant pressure than liquid water (that's why it floats to the top, after all), so it will need to displace the water. That's why this doesn't happen without the heating (and it will stop rather quickly after you turn the heat off, despite the fact that there's a lot of water ready to evaporate at the slightest disruption - that's why freshly boiled water seems so agitated when heated in a good electric kettle or a microwave).
As vapour forms, it causes the formation of a bubble - the surrounding liquid water will create a surface around the vapour, exerting a large amount of pressure. Most vapour bubbles never grow to visible size. They collapse back on themselves under the surface tension. And that's why the impurities and imperfections are important for bubble formation - they allow the bubble to survive long enough, because they present volume for the vapour to accumulate in without having to fight the surface tension. And the bigger the bubble is, the smaller the surface tension is compared to the pressure of the gas. Eventually, the buoyancy is large enough to dislodge from the surface and float upwards.
Of course, if you heat water fast enough, you can force visible bubbles to form even in a perfect kettle. Convection is really good at carrying the heat away, but you can pump more heat than convection can ever hope to carry. But at that point, much of the liquid becomes dangerously volatile, and you'd get lot of superheated water splashing around. The good old "heat water in a microwave and throw a coffee bean inside" trick uses this - there's enough energy in the water for a lot of evaporation, but no nucleation sites; then you throw in the bean (with its very irregular surface, and thus many nucleation sites) and you get huge amounts of bubbles at the same time, exploding out of the cup. Needless to say, these visible bubbles don't form at the bottom :)
In fact, you can clearly see this in a less explosive form when heating water on a stove. Just put a spoon inside the pot, and if the conditions are right, bubbles will start to form on the spoon - again, not just the bottom (and sides) of the pot. The water is hot enough to form vapour under the pressure, but not hot enough to form visible bubbles - until you add the nucleation sites.
