Pokémon Black and White/Route 3 and Wellspring Cave

Route 3
Right near the start of the route, you'll find some Twins that you can battle in your first Double Battle. They each have a Purrloin L10 and aren't too hard, but it's a good introduction to Double Battles.

There's another Trainer up by the buildings that has a Munna on her team, which will be your first exposure to the odd Psychic-type Pokemon. The three Kindergarteners she's watching over all have the elemental monkeys on their team, each at level 10. You can also find a hidden Ultra Ball if you search really hard in the sand; it's right around the middle, so search carefully if you want it, you'll want this if you are a dedicated trainer.

Are your Pokemon getting weak from the fighting? Well you can just take a nap in the kiddie day care, which is actually right next to the Pokemon Day Care. That'll restore your Pokemon's health right away. Speaking of the Pokemon Day Care, it's right next door, although you can only drop off one Pokemon there for the time being, so don't be thinking you'll walk home with a horde of eggs.

 How can I breed Pokemon at the Day-Care?  '''At first, you can only drop off one Pokemon at the Pokemon Day Care, but after you've been to Nimbasa City (where you get your 4th Gym Badge), you'll be able to return here and leave two Pokemon. If you leave a compatible male and female Pokemon here, you'll end up with a Pokemon Egg of the female's species after a little while! Two Pokemon are considered to be compatible if they have at least one Egg Group in common, which you can see in any of the Pokemon's Pokedex entries here on the site.'''

Anyway, as you head west, you'll be stopped by Cheren, who wants to battle you right this instant!

Rival: Cheren 🇨🇴

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🇨🇴 🇨🇴

After defeating Cheren in battle, you'll see two Team Plasma Grunts run to the west of Route 3. Bianca and a young girl will follow behind, chasing the grunts. It appears that the Grunts have stole a Pokémon from the girl. Chase the Team Plasma Grunts with Cheren.

As you head west you'll come upon the entrance to Wellspring Cave. According to Cheren the Grunts have fled into the cave. Enter the cave, but be prepared to have some battles.