After you pass through the little building to the west of Accumula Town, you'll get a call from your Mom on your XTransceiver. After she hangs up, she'll arrive in Route 2. She gives you the Running Shoes, which, as in previous games, you can use by holding the B button to run. Your Mom will then return home.
There are some trainers here in Route 2. You also have a chance to catch a Purrloin in the grass, although it's not as common as the Patrats and Lillipups. There are also some items in Route 2.
Once you make it to the end of Route 2, you'll encounter Bianca, who challenges you to a battle. She will have a Lillipup, as well as the starter that she chose, which is the one that is weak to your starter. Once you defeat her, you can move on to Striaton City.