You can use the upstairs portion of the Pokemon Center now, so if you'd like to use the Union Room, Wi-Fi Connection, or Global Terminal, go for it. The GTS has a new feature that's worth checking out too, called GTS Negotiations, where you're matched up with random players and you can negotiate trades based on what Pokemon you each have.
Aside from the major locations of note within Nacrene City, there's also a girl in the house to the west of the Pokemon Center than will trade you a Petilil for a Cottonee or vice versa depending on the version of the game you have.
To prep for you gym battle, it's worth it to head west into the outer portion of Pinwheel Forest for a while to snag some Fighting-type Pokemon and level up your current team a bit. The woman standing near the border of Nacrene and Pinwheel has TM94 Rock Smash.
Pinwheel Forest (outer area)
Wild Pokemon (~level 12-17):
Tympole, Pidove, Timburr
Audino (shaking grass)
Sawk (common in Black; shaking patches in White extremely rarely)
Throh (common in White; shaking patches in Black extremely rarely)
For all intents and purposes, it's probably easiest to just consider Sawk a Black exclusive and Throh a White exclusive, and just trade for one or the other depending on the version you have. Unless you live in a remote area with no Wi-Fi access, it's not worth the hassle of trying to find the one you need in the shaking grass patches (even among shaking grass patches, they only appear about 5% of the time – it's not worth the tedious search).
When you're ready, you can head back to the gym inside the Nacrene Museum and face Lenora, the Normal-type gym leader. Afterwards, more scenes to watch, and you get the Dowsing MCHN.
Pinwheel Forest (inner area)
Wild Pokemon (~level 15-17 normal grass, 22-25 doubles grass):
Normal grass: Sewaddle, Pidove, Venipede
Doubles grass: Swadloon, Tranquill, Whirlipede
Shaking patches: Audino (common), Pansage, Pansear, Panpour (rare)
Black only: Cottonee (normal grass, doubles grass), Whimsicott (shaking patches)
White only: Petilil (normal grass, doubles grass), Lilligant (shaking patches)
Not much to discuss here. Make your way through the maze, fighting Team Plasma trainers all the way through. Have the nurse standing just outside the inner area of the forest heal you whenever you need it. You'll notice a moss rock too, at the beginning of the winding road through the forest – that should come in handy post-game for evolving Eevee into Leafeon.
Pinwheel Forest is also a great place to use your new Dowsing MCHN. You can have more than one item Y registered now too, making it easier to pull up lesser-used key items whenever you want.
Once you've taken in the majesty of Skyarrow Bridge, your next stop is Unova's biggest urban center, Castelia City. Of course, the gym leader, Bug-type aficionado Burgh, isn't quite ready for you yet, so once you've dropped by the gym, head over to the dock just west of the Pokemon Center to deal with another Team Plasma situation. Follow Burgh back toward the gym and you'll see some Team Plasma goons in front of the building just north and opposite the gym. You know what you must do.
Burgh and his trainers are pushovers, so after the gym battle make sure you explore the rest of Castelia if you haven't already. The dock here is where you can set sail to Victory Island to obtain your Victini if you have the event-only Liberty Ticket too. When you're ready, head north to Route 4.
The perpetual sandstorm on Route 4 is a bummer, since your Pokemon will fatigue much faster with the constant buffeting, but at least there is a prefab house slightly northwest as your make your way through the route where you can heal your Pokemon.
If you feel up to it, you can also check out the Desert Resort west of Route 4 before making your way to Nimbasa City. Lots of trainers and wild Pokemon should go far to beef up your team if you feel a bit underleveled.
You can't go all the way inside Relic Castle yet, but if you decide to check it out, make sure you pick up one of two fossils from the backpacker near the entrance. The Cover Fossil will revive into a Tirtouga and the Plume Fossil gives you an Archen. You could trudge back to Nacrene now to revive your fossils if you're a huge fan of fossil Pokemon, but it's easier to just wait until you get HM02 Fly and take care of all your backtracking then.
As you enter Nimbasa City you're greeted with another Team Plasma battle, of course – this time they're harassing the poor old Day Care man. He rewards you for protecting him by bestowing you with a bicycle. Hooray! Y register that sweet ride immediately!
Nimbasa City is full of flash and glamour, and there are lots of attractions to check out, including the Battle Subway. One of the cashiers at the PokeMart inside the Pokemon Center sells a bunch of expensive TMs, so that's worth checking out to fill out your growing collection. Around the corner, in the building directly north of the Battle Institute, you'll find an NPC that gives you HM04 Strength too.
Before you head to the gym, stop by the Musical Theater (sigh) and talk to Bianca briefly (don't worry, you don’t actually have to participate in a musical if you don't want to). After that's out of the way, you're free to head southeast to the amusement park area where the gym is located. After an intense scene with N, you're free to face Elesa, the Electric-type gym leader of Nimbasa.
Wild Pokemon (~level 19-22 normal grass, 22-25 doubles grass):
Normal/doubles grass: Minccino, Liepard, Trubbish
Shaking patches: Audino (common), Emolga, Cinccino (rare)
Black only: Gothita (normal/doubles grass)
White only: Solosis (normal/doubles grass)
Once you've earned your Bolt Badge and you're done exploring Nimbasa, you can head northwest to Route 5, where you meet the League Champion for the first time. After a scene (Cheren is kind of a jerk, isn't he?) you can proceed across the Driftveil Drawbridge toward Driftveil City.
On Driftveil Bridge you can battle and catch wild Ducklett if you fancy, as well as pick up a variety of Wings, like Pretty Wing, Swift Wing, Health Wing, Clever Wing, Genius Wing and Resist Wing. Wings for everyone, really.
The most notable spots in Driftveil City proper are the Driftveil Market (stock up on MooMoo Milk!) and the Move Tutor's House, where each starter Pokemon can learn one of the Pledge moves, Water Pledge, Fire Pledge and Grass Pledge, which work in conjunction during doubles and triples battles.
Of course, before heading to the gym, you must dispatch the local Team Plasma scourge, so head south toward Cold Storage.
Cold Storage (outer area)
Wild Pokemon (~level 20-23 normal grass, 24-27 doubles grass):
Normal/doubles grass: Herdier, Vanillite, Timburr, Minccino
Shaking patches: Audino (common), Cinccino (rare), Stoutland (rare)
You know the drill by now. Make your way through the icy floors of the Cold Storage Building, battling Team Plasma grunts as you go. Once you've vanquished them all, Driftveil Gym Leader Clay will thank you in his own way and invite you to challenge him at his gym in the northeast corner of town.
This Ground-type gym is one of the easiest yet if you chose either Snivy or Oshawott as your starter, so power through to get your Quake badge. The elevator arrangement in the gym can be a bit confusing though. Once you've claimed your victory, Clay, being the unreliable cowboy that he is, admits that he misplaced the TM he was planning to give you, so head out to meet him on Route 6 and pick it up.
But wait – Bianca stops you as head toward Route 6 to give you HM02 Fly. Can you feel the sky opening up to your whim? Clay can wait if you feel like backtracking to take care of anything else before proceeding. The Day Care accepts up to two Pokemon now, so if you want to go back and breed your starter for trading purposes, now is a good time. You could also go back to Nacrene and revive that fossil if it doth please you, or go show Professor Juniper your Pokedex in Nuvema. Or would you rather revisit Nimbasa and take a ride on the Battle Subway? The sky isn't even the limit now that you have Fly, so spread your wings… and Fly.
Wild Pokemon (~level 22-25 normal grass, 26-29 doubles grass):
Normal/doubles grass: Deerling (color varies by season), Karrablast, Foongus (also appears as fake Poke Ball item), Swadloon, Tranquill (Spring, Summer, Fall only), Vanillite (Winter only)
Shaking patches: Audino (common), Emolga, Unfezant (rare), Leavanny (rare)
When you make it through Route 6 to Chargestone Cave, the entrance to the cave is blocked with Galvantula webs, but Clay shows up to clear a path and give you TM78 Bulldoze as promised. Before heading into the cave, make sure you use the healing house nearby to refresh your party.
This is your first fairly large cave to traverse, and it has quite a few paths to explore as you track down Team Plasma. There's one doctor trainer about halfway through that will heal your Pokemon whenever you ask, but don't be afraid to use a Repel or two if your Pokemon are running low on PP. Once you make it through to the other side you'll be in Mistralton City.
Part one - Nuvema Town, Route 1, Accumula Town, Route 2, Striation City (Tri Badge), Dreamyard
Part two - Nacrene City (Basic Badge), Pinwheel Forest, Castelia City (Insect Badge), Route 4, Desert Resort, Nimbasa City (Jolt Badge), Route 5, Driftveil City (Quake Badge), Cold Storage, Route 6, Chargestone Cave
Part three - Mistralton City (Jet Badge), Celestial Tower, Route 7, Legendaries: Cobalion and Virizion, Route 17, 18 and P2 Lab
Part four - Twist Mountain, Icirrus City (Freeze Badge), Dragonspiral Tower, Relic Castle, Route 8, Tubeline Bridge, Route 9, Opelucid City (Legend Badge), Legendaries: Tornadus/Thundurus, Route 10, Victory Road, Pokemon League