All three ways you've mentioned have advantages and disadvantages.
- MIC: If you're not in a loud band, you should be fine with a mic. Mics get more accurate sounds, but it's hard to keep other sounds out of them, especially in a noisy environment. For an unrambuctious acoustic show, they're good.
- AMP: Going into your own amp is easy for you to control. But it's hard for the person running sound to work with. Worst case: The show will sound bad, and you'll make an enemy of the sound person.
- PA: Going directly into a PA means giving up most of your control of your sound. You can tell the sound person what you want, but you won't always get it. When the person running sound is good: no problem. When the person running sound is bad: big problem!
You didn't mention the fourth option: Going into a PA via a preamp. That's what I usually do.
- PREAMP INTO PA: It gives you a lot more control over your own sound - without taking making life difficult for the sound person. The downsides: Obviously, there's the expense of buying one (and a pickup, if your guitar doesn't have one). And preamps can be tricky, therefore annoying, frustrating, or distracting. I went through a some utterly bewildering contraptions before finding one that (a) is easy to use and (b) sounds good.
In general, if your guitar has an active pickup, you'll want a passive preamp (or passive DI box), and If your guitar has a passive pickup, you'll want an active preamp. Either go active-passive-PA or passive-active-PA.