The moves you teach it may largely depend on what ability your Skitty has;

Normalize

This ability will give all of Skitty's attacks Same-type attack bonus, so priority moves like Sucker Punch (must be bred in) can get a little extra power behind their speed. It will also let Skitty use moves like Thunder Wave (TM73) on the otherwise immune ground types.

A downside to normalize is that ghost types will be immune to all your direct damage - this can be mitigated using Foresight (learned at lvl4). Additionally, Rock and Steel types will be highly resistant to your attacks.

Cute Charm or Wonder Skin (hidden ability)

Zen Headbutt (must be bred in) is a good counter to normal types' greatest weakness - fighting. High priority moves like the previously mentioned Sucker Punch or Fake Out (move relearner) are still a good choice.

Skitty can also learn Baton Pass and Wish though breeding - combined with Calm Mind (TM04), this can make for a fairly decent support (combine those three with a special attack like Ice Beam (TM13) or Thunderbolt (TM24) to let skitty take advantage of its own stat boosts.

Smogon University goes into further details on specific builds, giving suggested EVs, natures, and items as well as strategies. You can copy those movesets to the letter, or modify them to your liking.

EDIT: If you are looking solely to teach moves to an existing Skitty, and not do any fancy breeding;

• Swagger (TM87) and Psych Up (TM77) have an interesting, if risky, dynamic - Confuse the target, raise its attack two stages, and then copy the attack up buff. To make use of this buff, Return (TM27) is a good STAB move that has a base power of 102 at maximum friendship, and round it out with Faint Attack (lvl 29) for a little type coverage.
• You could try for Sing (lvl 11), Dream Eater (TM85), and Wakeup Slap (lvl 32) - but sing's low accuracy makes this a risky strategy.
• Toxic (TM06) - if you aren't fighting a steel or poison type this strategy can work wonders, poison them with toxic and then stall - Protect (TM17) Double Team (TM32), and for a fourth move Attract (TM45) or Swagger
• Para-fusion - the idea here is to minimise the enemy's chance to do anything useful and hit themselves Thunder Wave (TM73) and Swagger should set it up nicely - The danger to using swagger (particularly after several applications to keep the enemy confused) is that if they manage to get by both statuses (37.5%, I believe) and actually hit you it will HURT - You might consider Double Team to make yourself even harder to hit.
• Heal Bell (lvl 39) could be worked in to many of the above possibilities - it's a nice team support ability to clear the rest of your team's status ailments - particularly useful during some of the longer routes, means you don't have to be too vigilant about stocking up on the various curative items that may be in short supply and on the pricy side. (although by lvl 39 the cost and availability of the items may no longer be an issue)
• If you go for Calm Mind to buff your SpAtk, Echoed Voice's (TM49) power increases by 40 with each application (up to 200), and it's normal, so STAB bonus applies. If your Skitty is durable enough to survive a few hits it could build up quite nicely against trainers with more than two pokemon.

And if this is just against the NPC's throughout the story, always remember that there's nothing a little back-tracking and grinding out a few levels won't cure.