Amplifier design is a huge topic which would take pages (or a book) to adequately cover. I would recommend a Google search for that one.
Generally a kit from a reputable manufacturer will be fine, but really the golden age of good amplifier kits from the likes of Heathkit, Eico or Fisher are long gone. Offhand I couldn't name a current amplifier kit manufacturer, but then I haven't looked for one recently either. You probably won't save much money building a modern kit; its more about the fun of building than saving money.
Unless you have very efficient speakers (over 90dB), you probably don't want a tube amplifier. Efficient speakers act as a magnifying lens for very low level abnormalities in an amplifier like crossover distortion and high order distortion in general. Tube amplifiers have much less (or no) crossover distortion and their distortion components are generally lower orders (2nd, 3rd, 4th, 5th in descending order). Single ended triode amplifiers generally have a strong 2nd harmonic (which is benign since it is a musically related octave), a much lower 3rd, and no crossover distortion since there is only one output tube. Also, single ended triode amplifiers refuse to go into hard clipping. The problem is that they only have about 7 watts of power. Push pull tube amplifiers generally top out in the <75 watt range unless you pay very big bucks.
Solid state amplifiers of course are way less expensive and can easily have hundreds of watts. The problem is that they have no soul, but that's just my opinion!