The discussion of nucleation sites is very much to the point. Water at atmospheric pressure without nucleation sites will theoretically boil only at $320.7 {}^o C$. The bubbles act as nucleation sites that reduce the energy required for evaporation. In the case of a bubble, the effective contact angle between the superheated liquid and the bubble surface is $180 {}^o$ which reduces the superheat needed to evaporate the water to $0$. Interestingly, there is actually an impediment to bubble growth caused by the reduced temperature of the vapor inside the bubble and a corresponding lower superheat boundary layer of liquid surrounding it.
FYI: My information is based on Collier and Thome pages 138 and 549.
In that text, an equation is given for the rate of bubble growth as:
$$ R = \sqrt{\frac{12\alpha_f}{\pi}} \frac{\rho_f c_{pf} \Delta T}{\rho_g i_{fg}}\mbox{Sn} t^{1/2}$$
where
$$\mbox{Sn} = \left[ 1-(y-x)\sqrt{\frac{\alpha_f}{D}}\left(\frac{c_{pf}}{i_{fg}}\right)\left(\frac{\partial T}{\partial x}\right)_p\right]^{-1}$$
and the variables are:
$R$ - rate of bubble growth
$\alpha_f$ - thermal diffusivity of liquid
$\rho_f$ - density of liquid
$c_{pf}$ - specific heat of liquid phase
$\Delta T$ - temperature difference
$\rho_g$ - gas density
$i_{fg}$ - latent heat of vaporization
$t$ - time
$D$ - molar diffusion coefficient
It's been a while since I looked at this in depth, but I think the $x$ and $y$ variables refer to position relative to a uniformly heated tube coaxial to the $y$ axis. Honestly, I don't expect you to actually use this formula, but hopefully it will impress that there are people who have spent a great deal of time on this subject. If you find it interesting, you might have a promising career in power plant boiler engineering in general or nuclear power plant engineering in particular.
The bubbles are already on the surface, they are just too small to see with the naked eye.
Wetting a surface, even at room temperature, results in tiny gas/vapor bubbles at defect sites due to surface tension. For example, surface tension prevents water from seeping into tiny crevices (on the order of microns).
These tiny gas pockets expand when heated, and eventually you can see them. They were on the surface the entire time, they just expanded. They stay on the surface because surface tension pulls down and balances the upward buoyant force.
If you keep adding more energy, however, the gas in the bubble will expand. Eventually the bubble will eject from the surface because the surface tension scales inversely with bubble radius, so the force holding it back decreases. Furthermore, as the bubble increases in volume at the surface, it gains an appreciable buoyant force that overcomes surface tension. At this point, the bubble rises.
You can actually superheat water above the boiling point if you have a surface that has small enough defects, since this makes it more difficult for gas bubbles to be trapped when the surface is wetted.
Anyways, the bubbles seem to stick to the sides of the container because they were always there to begin with, thanks to surface tension. You only see them when higher temperatures cause the gas inside them to expand.
Best Answer
the "early" bubbles which form before the kettle begins to boil are bubbles of air which come out of solution because air is less soluble in hot water than in cold water.
Those bubbles are very small in comparison to the viscosity of the water, which means they are easily moved about by convection currents in the water, and they rise slowly.
Now, note that the top-most part of the kettle is cooler than the bottom-most part, where the heat is being added. So air comes out of solution in the hottest part of the kettle, is carried around by the currents, and rises slowly. In so doing those bubbles encounter cooler water and redissolve before they reach the water surface.
Hot water under pressure in pipes can hold more dissolved air than at atmospheric pressure. Once the hot water exits the faucet valve, its pressure drops to atmospheric and the dissolved air tends to come out of solution and produce a milky/cloudy suspension of very small bubbles in the water. This process depends on how much air was in the water to begin with and how much time the water spent in the water heater, getting degassed.
Cold water can contain significant amounts of dissolved air and does not become supersaturated when dropped to atmospheric pressure upon exiting the faucet.