Battle Moveset

A Good Moveset for Latios would have to be this:

Attacks:

Dragon Claw
Psychic
Thunderbolt OR Calm Mind
Ice Beam

Items Attached:

Soul Dew

Preferred Nature:

Timid

Strategy Using Latios

Latios is quite a strong pokemon due to it having a fair spread of stats, with its defense being the only one slightly below average. It has diverse movepool too which allows it to learn many moves to support those awesome special stats.
If you want, you could put Calm mind on there somewhere, just to give a boost to its already high special stats. Anyways, as a pokemon, its been given a decent type combination, and doesn't have the 4x weakness to ice that a majority of Dragon pokemon do.
Although Dragon claw, Psychic, Thunderbolt and Ice beam can deal considerable damage to most pokemon. Both Dragon claw and Psychic get STAB making them dangerous moves to go up against. Since most pokemon that learn Ice beam are water pokemon, Thunderbolt takes care of them.
If you don't want Soul dew on Latios, replace it with either a Petaya berry or Salac berry. Max out EV's into Special attack and speed to get those extra stat boosts that is useful for Latios, especially with this moveset.
Latios is flawed by one thing, its classification of a legendary, and with such a title, its become considered uber and overused. With high specials, attack this pokemon using Physical attacks as its defense is the lowest of the two defense stats.

EV Corner:

Max out the EVs in Special Attack & Speed for Latios with any remaining in Special Defense

Strategy Against Latios

Latios's Defenses are both high but it's Physical Defense is lower so attack with a volley of Physical Attacks or Dark, Dragon or Ice Attacks and it'll be gone in no time

Contest Moveset

A good Contest Moveset for Latios would have to be this for the Cool Contest best with Lonely, Adament, Naughty or Brave Nature:

Attacks:

Thunder
Thunderbolt
Rain Dance
Hyper Beam

Items Attached:

Red Scarf

Strategy Using Latios

If you do the attacks in this following order you should have very few problems; 1st - Rain dance, 2nd - Thunder, 3rd - Rain dance, 4th - Thunderbolt/thunder, 5th - Hyperbeam

Locations in Games

Ruby/Sapphire Random Grass & Waters in Hoenn after beating Elite Four Colosseum Trade from R/S Fire Red/Leaf Green Trade from R/S

Animé Appearences

Latios has had a single Animé Appearence. In it, it was in Alto Mare and had to protect the Soul Dew and Alto Mare from Annie & Oakley. It ended up sacrificing itself to protect the city Movie 5: Heroes - Latios & Latias