Best tactics for Gyms.....

For prestige the best way to go is to chose an attacker with half CP of the first defender in the gym. Whether you choose just the first defender or go for the next ones will depend on what defenders are there and what CP they have. Say if you see a Kingler or Pidgeot in the first slot and Snorlax or Lapras up next you would probably want to just work on the first one... if there are several weaker defenders it makes sense to work on 2 or 3 in a row to optimize prestige gain. Don't worry if you don't get 1000 points from the first one, if you take out 3 you will still get good prestige.

As to enemy gyms - just pick your best team of attackers :)

First thing I do is set myself a Goal based on where Pokemon are.

Snorlax and Lapras are often a stopping point. Unless there's a couple Gyrados right after :>

Then I pick my team to beat that Goal. Usually my team is around 1400-1500. And I am beating pokemon in Gyms from 1800 at bottom to 2400 - 2700 range depending on Stop Point.

I tend to aim for at least 2k Prestige a Run.

Also remember, gym snipers are watching. If you are prestiging, keep track of exactly how much you need to prestige to bump the gym level, then run and place your defender. The gym updates with every defender you defeat, so if you accidentally level up the gym, but keep attacking another defender with your six attackers, the spot you created could be gone before you even notice. Other than that, I shoot for the max I can get per prestige attack with the 1/2 CP attackers. Had a nice 4K per attack last gym, so needing 7K, I defeated four defenders, then three, then placed.

First, you should analize the gym you are targeting:
For attack, just use the best attackers you have, preferably ones with type advantages.
Prestige:
- How much prestige do you need to drop a pokemon?
- What's the CP of the lowest pokemon?
- Is there a snorlax or lapras in the gym?
- Where exactly are said lapras/snorlax?

With that information, you want to get at leas 500 CP per pokemon defeated, you get 500 if all of your team has lower CP tha the pokemon you defeat, as the max CP in your team decreases, the prestige gain increases up to 1000 if your team's max CP is half of the CP of the defending pokemon or less.

This means that If you generally want the CP of your team to be as low as posible to maximize prestige gain... as long as you can consistently defeat the defenders in the gym, for this, you use type advantages and dodging.

Now you can find the following problems:
- The CP of the defenders is really low or there's a really low CP one at the bottom
- You find a snorlax or lapras, that are a nightmare to take down.

You have a few options, in case there's a gap between some of the defenders, you can either focus on the ones before the gap, or the ones after the gap, meaning that you either put all of your team below the CP of the pokemon before the CP gap, or you ignore the ones before (they should be easier to take down), and focus on the ones after the gap.

As for snorlax or lapras, I only prestige against them if I have no other choice (they are the first pokemon in the gym).

You don't have to allways use a team of half the CP of the defender, 500+ prestige per pokemon is a good amount, specially if that means you can fight 4 or 5 with a team of 6

To add to the info already mentioned, when attacking you don't need to finish the gym. You can choose to steamroll the first few easy ones and run a few times to make the process quicker.

Good for avoiding the upper rungs of time sucking snorlaxes until needed.

Since everyone is giving advice on prestiging, I'm gonna give a tip for attacking lvl 3 gym (which are by far the most common level around my town).

If enemy gym is level 3 and it has 6000 prestige points, instead of battling all of the Pokemon, first kill 2 bottom Pokemon and then run away (2000 prestige), then attack again and kill everything (3000+1000) and the gym is done. This way you will have to fight the highest CP defender only once instead of having to battle it twice, while the total amount of fights stays the same (5).

So instead of this:
3 > 2 > 1 ----> 2 > 1

This:
3 > 2 ----> 3 > 2 > 1

Taking down a gym is simple. Bring your biggest baddest Pokemon and beat as many (hopefully all) of the opposing gym Pokemon as possible. You get an extra 1000 prestige knocked off if you beat them all at once.
Building up a gym you need to really pay attention to typing. Bring the lowest (up to half of enemy CP) pokemon you can. Beat as many of the pokemon in the gym as you are comfortable with. Sometimes its worth only beating the first few if there is a rather large CP discrepancy between the defenders CP.

With all the good advices above, I think now you know about the game mechanics.
My advice is: KNOW YOURSELF and KNOW YOUR CUSTOMERS :D
- Knowing what you have in your hands: your pokemons' CP, their type advantages, their movesets... You only need around 10 good defensive, high CP pokemons. For offensive purpose, only keep the best attacking mons of each type (water, grass...) with wide range of CP for both upgrading friendly gyms and taking down other team's gyms.
- Know your customers (especially when prestiging) so you know when to take down one lowest pokemon and run away, when to skip the first pokemon (attacking it with higher CP pokemons) to aim at higher pokemons in gym but have many disadvantages against your attacking team.
- When fighting, pay attention to the moves of defenders (mostly charge moves): one bar charge moves are easy to dodge, 3-4 bar charge moves are usually weak, 2 bar charge moves are the hardest (body slam, dragon pulse, aqua tail...). It's usually safe to dodge the defender's charge then use your charge move as some charge moves have long cooldown times that you can't have enough time to dodge.