Your mom will be outside of your house and will have received a call from Professor Juniper essentially telling you that you'll be able to pick up your first Pokemon from her assistant, Bianca, so she goes inside to let you know. Walk on over to her and she'll ask you a few questions about how to use the menu; just answer yes to each one and move along out the door.
Go on and head north and you'll be stopped by Hugh and his little sister. They'll say that they saw Bianca up by the lookout in the north end of town. Hugh will follow behind you afterwards, so just go take him north, behind the houses and the Pokemon Center until you see stairs leading up. Take those stairs up towards Bianca, although Hugh will decide to stay back to let you talk to her first.
Go ahead and talk to Bianca and she'll introduce herself, asking if you're (your name). Well, quite obviously you are, so answer yes. She seems to have taken Cheren's glasses from the previous game and has also changed her attire a bit. Anyway, she'll then ask if you'd like to take a Pokedex, so just answer yes.
After answering yes, you'll be given a brief moment to, say, save your game before you choose your starting Pokemon. Do that if you want or if you are going for a specific nature. Whenever you're ready, talk to her again and she'll give you the option of picking one of three starter Pokemon: Snivy, Tepig, and Oshawott.
After choosing which Pokemon you want, you'll be able to give it a nickname, should you so desire. Bianca will also give you the Pokedex after you've decided whether or not to nickname your Pokemon.
Once you've gotten that, head on south and Hugh will show up and talk for a bit. He seems like quite a tough guy. As you walk away, though, you'll have to fight Hugh! He'll use the Pokemon he hatched from an egg against you.
(if you picked Snivy)
|(if you picked Tepig)|
|(if you picked Oshawott)|
There's very little strategy to this battle. No matter which Pokemon either of you have, it only has Tackle and a Defense-lowering attack, so the fight will be over in three turns, barring a critical hit. If you win, you'll gain $500 and enough EXP to gain a level up to level 6, but if you lose, it's not the end of the world at all and nothing bad happens, other than you missing out on the money. Don't feel bad if you do lose, though.
After you beat him, he'll run off, and then Bianca will give you a tour of the Pokemon Center. You'll get your Pokemon healed up and everything. She'll then show you the PC. Woohoo, that's great. Hopefully you are familiar with the navigation of the box system. It pretty much the same as in the original Black and White.
Finally, she'll show you the Poke Mart inside of the Pokemon Center. She'll even give you an astonishing 10 Poke Balls to get you started. Nice!
When you leave the Pokemon Center, your mom and Hugh's sister will show up. She'll give you a pair of Running Shoes, which let you run by holding down B. Hugh's sister also gives you the Town Map, which you can use to look at the map of Unova. You're given an extra one to deliver to Hugh, too.
Anyway, whenever you're done, head on north and onto Route 19! As you head through the gate, though, the attendant there will give you a free Potion. Try to save this if you can and just return to the Pokemon Center if you need healing, but it's good to have this in case of emergencies.