Instructions
Raiko and Entei
Explore the burned tower in Ecruteak City in order for these two legendary Pokemon to escape and roam the area.
Place a Level 39 Pokemon with a move like "Mean Look" at the head of your party. Both Raiko and Entei are Level 40, which means you are able to use Repel to avoid encounters with weaker Pokemon while searching for them without repelling them as well.
Buy Repels and Ultra Balls from a PokeMart, then leave the town and walk along the routes until you randomly encounter Raiko or Entei. Use the Repel to cut down on random encounters with wild Pokemon, which will waste your time and slow down the search. Raiko and Entei can appear anywhere in the Johto region, but you can track them by looking at the map in your Pokegear. A small icon with their faces will show in the area where they are.
Use the "Mean Look" special move against the Pokemon as soon as you encounter them in battle. This will prevent them from running away and requiring you to chase them again. Use moves like "False Swipe" to reduce their health without killing them.
Throw the Ultra Balls you have bought at the legendary Pokemon once their health is in the red. Eventually you will catch them.
Latios
Travel to Saffron City and speak to the girl who lost her doll in Vermilion. She ask that you get it for her in exchange for a train ticket.
Travel to Vermilion City and enter the Pokemon Fan Club located there. Get the doll from the man inside the club, then exit. Once you are outside, a man will approach and inform you about Latios being loose in the world. From this point on, you will be able to catch Latios.
Open your map and look at the location of Latios; it is marked by an icon of Latio's face. Latios keeps changing locations, so if you travel to where it is, chances are it will be gone by the time you get there. Instead, enter and exit an area while checking your map until you encounter Latios in the area that you are.
Use a blocking move like "Mean Look" on Latios as soon as the battle starts to prevent it from fleeing. Attack it until its health is in the red, then throw Ultra Balls at it to catch it.