The first step to finding the Togepi Pet is to wait until Ash respawns in his hidden cave. He is a rare, very long spawn (several days). If he's there, make sure you don't have a pet with you already.
Once you have found ash and his chosen pokemon, you need to kill the pokemon until a Missingno spawns. You need to kill Missingno until you get a pokeball? drop from the Missingno.
Next, you need to weaken ash until he is very nearly dead, and use the pokeball? on him. If successful you will gain Ash's Backpack.
The next step is to head to Tetra (Yellow for Gold) and collect 850 Tetra Dust. This is dropped by Silphs. They drop 3 Tetra Dust about 84% of the time, Lustrous Shards about 14% of the time and 10k Gold Bars about 2% of the time.
Once you have your Tetra Dust, head to Lenarah to trade for a Bottle of Sand (key, dissolves).
Once you have your Bottle of Sand, head back to the Ipsen area and find the Quiz Trial Building with the fire beside it.
Use the Bottle of Sand on the fire (it will dissolve) and enter the building from the rear.
Head inside and talk to the shop keeper to trade your Ash's Backpack for a Mysterious Egg.
Simply use the Mysterious Egg to hate your brand new Togepi!