Extra Missions
After you've beaten Pokemon Ranger and the credits have rolled, your game will be saved. Load it as you normally would, and then you'll see a large cutscene, allowing you to eventually gain access to the data of your partner's Pokemon. Then, take the elevator to the second floor and talk to the man next to the Dragonite, who will allow you to choose what city you want to go to. Pick "F", for Fall city.
Fall City:
Try to enter the headquarters and you will receive a message. Now, go to the house where you used a Staryu to light the basement, and talk to the guy who is inside, but not without first releasing the Pokemon which you captured, if any. (In case you don't know how to do it, simply access the menu by pressing START, select the upper option and then select each Pokemon in there, pressing the upper button after each one)
You will get a new cutscene and eventually be back to the headquarters in this city and get a new mission, when you exit this headquarters.
Extra Mission 1
After the cutscene, go down the ladder and you'll have to fight 4 Staryu at the same time. Stun them with the electrical attack and then easily capture them. When you're back in the outside, visit that house you've just seen and then you will have to beat 4 Krabby, which you can easily capture (c'mon, they are really slow!). After such battle, go to the outside and then slightly down, where you will see yet another scene. Make your way to the Pokemon in the lower right corner and you will have to fight them. Stun them all with a fully powered electrical attack and you will then be able to capture them easily.
Now, you'll see some strange things in the sea, so you should board in the Lapras and you'll (eventually) be taken into some whirlpools, where you will get to see a Kyogre. He isn't strong, but he can summons waves that seem to cause massive damage, so you should wait until less waves are coming (that *eventually* happens, you just have to wait!) and then capture him as quickly as possible. The electrical attack may also slightly help you, but be sure to avoid the waves at all costs, as they may put you into trouble.
After the battle, you will be taken to the headquarters nearby and your mission will end.
Fall City:
After the long conversation, you will find yourself outside of the headquarters. Therefore, enter it once again and then take the elevator up, using the Dragonite to go to Ring Town.
Ring Town:
In here, enter the headquarters, release all the Pokemon you may have (if any) and then talk to your boss. After all the conversation, you will be taken to Summer Town.
Summer Town:
Some more conversation, an eartquake will occur and you will eventually be given a new mission. Exit the headquarters and then you will find yourself instantly at the fire cave, where you found that pyramid from before and your mission will start.
Extra Mission 2
Go to the right, up, cross the bridge to the left side, continue going and you will see that your partner is trapped inside that cave with a huge rock. Capture the Rhydon nearby and then use him to get rid of that huge rock. Small cutscene, enter the cave and be sure to capture the 2 Poliwrath.
Back to the outside, head all to the right across the bridge and use one of the Poliwrath to get rid of the huge glowing fire stone. Capture as many Pokemon as you want, but be sure to keep the other Poliwrath with you at all times. When you're ready, go all the way up and enter the next area.
Continue going up and fight the guy from Team GoGo, having some help from any fire Pokemon to easily capture the two enemies. After defeating them, head to the left side and capture (and use) the Manectric as many times as it takes to fully heal you. When you're done, capture him yet again and then go up the stairs on the right.
On the next room, go all the way down and down the stairs. Capture the other electrical Pokemon, another Manectric on the upper left corner and go up those stairs, on the upper left corner. Explore that room up to the end, as you can find some Charmeleon in there. You can also capture another Rhydon in there, if you want. After doing so, go to the outside and to the right, saving your game before going up the stairs.
On the next room, go all the way up until you reach a new room and you will be able to capture an Arcanine, which you can easily stun (with the eletrical attack) and capture. Now, go all the way down and back to the room with the 2 Charmeleon, where you should drop down an hole in the floor, which is placed between the 2 Charmander and the place where you could previously see 2 Charmeleon. You'll see a new scene and you should then enter the door that just opened.
You'll find a Groudon inside, which you can easily capture with the help of your Poliwrath. You kept it, didn't you? I hope so... After the cutscene you'll be brought back to town.
Summer Town:
The usual cutscene, and your mission will be (literally) over. When all the conversation ends, you should enter the headquarters once again, take the elevator up and talk to the guy near the Dragonite. This time, go to Win Town.
Win Town:
When you're there, talk to the woman in the right (not before releasing your Pokemon, as you did before) and a bigger cutscene will follow, where the other guy from before will join you and you will gain yet another new mission. Exit the headquarters and you will instantly find yourself in the place of the mission.
Extra Mission 3
Start by going up and you will have to fight a guy with 3 Ekans. Easily defeat them and then search the entire area for a Ralts, a Kirlia and a Gardevoir. I can't tell you where they are since they seem to change their location, but you can simply continue searching and eventually you will get all 3 of them.
After doing so, visit the place where you previously put the 3 evolutions of Eevee and you'll see a cutscene. Interact with the 3 stone blocks and put the 3 Pokemon I told you to capture in there, which will lead to yet another cutscene. Go up and fight 2 Pokemon in that huge building, which you can easily defeat by using your usual thundershock. Now, go all the way down to the previous area and capture the following Pokemon: Medicham, Gengar, Gengar (yes, 2 of them, once again...), Espeon, Umbreon, Raichu. Their locations are the same as before. Once you have all of them return to the huge building and fight the second group of enemies. Using a single Gengar, capture all of them and then continue going.
You'd better heal and save your game at this point, but it is really up to you. At the top you'll find a Rayquaza, which you should capture with some help from a Gengar, which will make this battle really easy.
After the battle, some more conversation will occur and you will be back to town.
Win Town:
A lot more conversation and then your mission will really be over. Some time later, you will finally see the (usual) screen that allows you to save the game and then it will be over. Congrats, you have now completed all the extra missions!