Cannot find Prof. Sycamore in Lumiose Station after defeating The Elite Four

In Dendemille town, in a House, up and right from the Pokemon Center (next to some stairs), is the Move Deleter and the Move Reminder. The Move Deleter is to the left side of the house after entering, and the reminder is in the right side of the house.

Just like the previous Pokemon games, the reminder will not remind your Pokemon for free, you need a Heart Scale, carried by Luvdiscs or found in Rock Smash rocks.

Either catch them and take the item from them (at a cost of 200 per Pokeball), or use item-stealing moves in battle such as Thief and Covet, and take them off your Pokemon after battle. Luvdiscs are found in bodies of water, while fishing with an old rod.

The other answers are pretty good, but just to add a couple more points.

• Work on filling out your Pokedex
  • Go back to different areas and catch Pokemon you're missing (check the Pokedex for their locations).
  • Trade between X & Y to get the ones you can't catch in your game
  • Evolve Pokemon you've caught already
  • I usually set up a Pokemon geared toward catching other Pokemon. Smeargle works well for this - Teach it False Swipe, Mean Look, a Paralysing move such as Stun Spore, and something else (I usually run with a move to hit Ghost Pokemon)
• Try your hand at multiplayer: Link battles tend to be a lot different to Singleplayer battles - they have their own tactics, and the battles tend to be harder as they're more evenly matched (where Singleplayer is geared towards you winning). You really have to focus on Team cohesion a lot more.
  • Check out Smogon for different tactics, and check which Pokemon work well (and work well together). It's still geared toward Gen 5, as the community is still learning the ins and outs of X/Y, but most of the info there will stay relevant.
  • There's a big focus on having Pokemon with the right Individual Values (IVs), Effort Values (EVs) and Natures. You can be average in multiplayer without worrying too much about these, but if you plan on entering tournaments this is where you'll gain the edge over your opponents.
• Trade for all the Vivillion variations: There are 18 different wing patterns, depending on which real-world Geographic area you hail from. See if you can trade for all types.
  • Most people on GTS will trade Scatterbug for Scatterbug, so this would be my first point of call.
  • Once you start looking for the harder to get ones, I'd check out some of the forums/reddit posts for people wanting to trade duplicates.