Part of her duties as the Slayer involve Buffy patrolling Sunnydale in the manner of a cop, Batman or similar. In the first two seasons, this was a very risky business for her. Not only was she relatively inexperienced (she was new to the area), but she had to make sure that no one saw her patrol. Her mother, in particular, had to be deceived - and Joyce helped out by embracing denial:

"How many times have you washed blood out of my clothes?"
- Buffy, "Becoming Pt 2."

In seasons three and four Joyce started to accept Buffy's calling, and so the Slayer didn't have to sneak out of her bedroom window anymore. However, that didn't take away all of her work. What actually goes on to make sure that a patrol happens like its supposed to?

Patrol usually takes place after sunset, continuing for a few hours. However, a sleep-deprived Slayer is a cranky Slayer - and we all know that a cranky Slayer is a careless Slayer. So one assumes that patrol doesn't take place after, say, 1am. In California (esp. during the summer) that means that Buffy is likely to patrol between 9pm - 1am (not necessarily for that long, though!).

'People' places are top of the list: that's where vamps would go to feed. That means places like the outside of the Bronze and the back alleys around it,

the streets around the coffee bar, the college campus in season four,

 

and the beach, if it's warm/someone wants to throw a party there anyway. The other big feeding grounds are the make-out spots dotted in and around Sunnydale. Those spots are tempting for *all* demons, not just vampires, because they are usually in secluded areas that are hard to access (like the woods). In fact, one of those areas was the target of a werewolf attack (along with the Bronze, which was also targeted for its high percentage of sweaty teenagers).

The other places regularly patrolled by Buffy are Sunnydale's 12 cemeteries. That is where the new vampires are likely to rise (along with the morgue, which is occasionally checked, and the odd rising in the funeral homes), so that is where the Slayer waits to pick them off while they are still young and vulnerable. Also, some vampires (most notably Spike) are partial to crypts for a home, and so are 'visited' by the Slayer on occasion.

Buffy's usually on her own when she patrols - at least, she started out that way. As the Scooby Gang quickly took 'Rambo' training, they began accompanying her on patrol (presumably to hold Buffy's mochaccino and croissant while she kicked demon butt). They eventually became accomplished enough to patrol without Buffy (working as a group) when she left in "Becoming Pt 2".

Willow: Still, we were getting good. We dusted (nods) nine out of ten.

Oz: (whispers to Willow) Six out of ten.

Willow: Six out of ten.
- Oz & Willow, "Dead Man's Party".

In Season 3 Buffy began patrolling with Faith - which didn't really work out well, because the sheer monotony of the routine got to the second Slayer. However, Buffy didn't go back to patrolling alone after Faith left the picture - in Season 4 she alternated with Willow and Riley's commandos - and even, one supposes, with Spike. Which we can assume made them both very happy.

Spike: I say we go out there (Rubs his hands together) and kick a little demon ass! (Xander and Willow stare at him) What, can't go without your Buffy, is that it? Too chicken? Let's find her! She is the Chosen One after all. Come on! Vampires! Grrr! Nasty! Let's annihilate them. For justice - and for - the safety of puppies - and Christmas, right? Let's *fight* that evil! - Let's *kill* something! (Fade to black) Oh, come *on*!
- Spike, "Doomed"

Buffy usually wears trousers and a jacket when patrolling. This ensures mobility and allows her to hide several weapons on her person.

Vampires turn to ash when staked, which is wonderful for removing the need for disposal of the bodies (unfortunately, if it rains the ash will stain). Oher demons, however, aren't that considerate. On more than one occasion we've seen Cordelia and Doyle dismember and dispose of a body for Angel - one can only assume that Buffy is no stranger to that either. Which brings up the burning question - does the Slayer carry rubber gloves in her pockets?

Cordy: “Oh. That guy’s never going to put himself back together, right?”

Angel: “He’s in 12 steel boxes buried in 20 cubic feet of concrete in the floor of LA’s newest subway station.”
- Cordelia & Angel, "Pieces".