Quick answer: a highball glass for fizzy drinks, a rocks glass for spirits over ice, a coupe or martini glass for 'no ice' cocktails (served chilled after shaking/stirring), a shot glass for layered shots. A garnish should come from the drink's ingredients or complement them aromatically.
The glass affects the physics of the drink: a narrow tall highball holds carbonation longer, a wide rocks glass lets ice slowly open up a spirit, and a coupe's thin stem keeps your hand from warming the drink.
Chilling the glass is a must for cocktails served without ice: fill it with ice and water while you mix, dump it before pouring. A warm glass kills a Martini or a sour within a minute.
Citrus peel is the most effective garnish: squeeze a strip of peel over the glass, skin side down — the oils settle on the surface and deliver aroma from the first sip. Then run it around the rim.
Room for garnish: mint (a sprig, lightly slapped between your palms to release aroma), berries on a pick, a citrus wheel on the rim. One rule: the garnish should smell like something that's in the drink — otherwise it disorients.
The final touch — a napkin or coaster under the glass, and serve immediately. A cocktail lives 10-15 minutes: after that the ice dilutes, the fizz fades, the aroma is gone.