I'm joining a new campaign soon and I have an idea that I want to play a Changling who likes playing tricks but I can't decide what a good class for that would be. I'm thinking of a spellcaster like maybe Cleric or Paladin.

You answered your own question.  Trickster changling.  I.e. an Arcane Trickster rogue.  Expertise in Deception, Stealth, and/or Sleight of Hand combined with an invisible Mage Hand and a small but useful stable of illusion and enchantment spells.  

I'll give another possibility: Changeling Warlock. With Pact of the Blade, you'll be able to not only change your physical appearance but will also be able to summon a different weapon as an action to further diversify what you look like. 

You'll have good Charisma for Deception and Persuasion checks (and weapon attacks if you choose the Hexblade Patron), a magic weapon at early levels so you don't have to rely on the DM granting you magic items, spellcasting up to 9th level spells, the versatility granted by regaining spell slots on a short rest, and the roleplaying potential granted by having a Warlock Patron. 

If you really want to lean into the "tricksy" theme, choose the Archfey or Genie Patron, but I personally recommend the Hexblade. 

Clerics have the rather obvious choice, the Trickery domain.  Pick up a caster feat or two if you need some spells not on your list.

Paladins, good bad or ugly, tend to be pretty serious about whatever thier deal is, so unless you're willing to go full Star Trek & explore spaces no one has explored before, I suggest passing on them.  Plus if your tricks go bad, the only way out for a Paladin is likely to be violence.  So either you're not being tricksy (boring), or you're constantly dragging the entire party into possibly very inconvenient & unwanted fights (rude & inconsiderate in and out of game). So reason two to stay with a character that can either not be suspected of shenanigans, or get itself out of the kitchen when things get too hot without involving everyone else.

Bard could also make a good choice if Yurei's suggestion of Arcane Trickster isn't quite arcane enough for you.  Either way, enchantment & illusion are on brand for your concept.

Don't forget to check out magic items you'd prefer.  Anyone can swing a sword +1, but a Hat of Disguise is where the real fun starts.

Trickery domain cleric is a solid choice for a trickster, as is an arcane trickster rogue. The most memorable experience I had with a trickster changeling was a College of Valor bard who alternately masqueraded as a cleric of Lathander, a ranger, a College of Lore bard from a rival faction to his true faction and probably a host of others that I'm forgetting. This particular character had the actor feat, granting advantage on those deception and performance checks for impersonating others. I tend to think this particular character had the faction agent background (you can probably guess which faction since it was Forgotten Realms), but you wouldn't go wrong with the criminal, urchin or charlatan background to play a trickster. An illusionist or enchanter wizard could also be a good trickster, depending on what you want your tricks to look like, but could be difficult if you don't have enough good rolls to spare one for charisma, or are using point buy.

I'm surprised no one has said bard. College of Lore, Creation, or Whispers all have good options mechanically and good flavor thematically.

Okay, here's my hot take: The best changeling is not a changeling.

If I were you I would start with Variant Human and pick up the Actor feat. Then pick Warlock with the Archfey patron and make sure you get the disguise self spell (Trade it out for something else at level 2). At level 2 get the Mask of Many Faces warlock invocation and boom. You have the best low level changeling you could ask for. 

Mask of Many Faces gives you disguise self at will. Disguise self is non-concentration meaning you can have it up with no side effects for ever so long as you chain cast it. 
Actor gives you a bonus to charisma which is very nice at first level as well as advantage on persuasion and deception checks to pass off as another person....you see where I am going with this....
Warlock gives you access to spells and other invocations as well as all the Charisma skills to pick from. Archfey in particular gives you access to Charm abilities and teleports as well as superb low level spells like Sleep and Calm emotions.