I'm building a Halfling Rogue for a new campaign. She has equally high dex and charisma, and likes to perform magician style magic tricks with slight of hand. I'm of course going the arcane trickster route, but I'd also like to multiclass into Bard. Looking for advice about how many levels of each to take and when to take them, especially since this is my first time multiclassing. Thank you!

Str 8; Dex 16; Con 13; Int 10; Wis 12;  Cha 16

Additionally, if anyone has roleplay ideas for using the magic tricks, I'd love to hear them. For example I had the idea of her pulling a copper piece from behind someone's ear and giving it to them for "good luck" aka bardic inspiration.

You may want to consider Swashbuckler roguish archetype, instead.  Trickster uses Intelligence as its casting stat, which clashes with your existing stats, and Bard levels kinda cover low-level magical tomfoolery.  Swash rogue, on the other hand, is driven by your Charisma and very much plays into the dashing sleight-of-handy charismatic swordsman thing.

As for How Many/When?  That's entirely up to what level you figure the game will be played to and what you want to emphasize.  If you want to emphasize roguish talents with a dash of performance flair and bardic enhancement, I would go no higher than three levels of bard.  If you want to emphasize bardic magic with a dash of roguish talents and criminal seasoning for some extra bite, I would go no higher than three levels of rogue.  You generally want at least two levels of your 'main' class for each level of your off class, but every game is different and every character will have different needs. 

I would honestly recommend starting as a first level rogue, taking your second level in Bard for Rogue 1/Bard 1, and then approach each new level with the question "if the game died before I got another level-up, what would I want my last level on this character to be?"  Get your roguish basics and your bardic basics AQAP, play the character that way, and decide as you go which feels more fun to emphasize.  The particulars of your campaign and your party will be far more important than the advice of Internet randos in this case.

I didn't realize Arcane Trickster was intelligence based, thanks for bringing that up! This was exactly what I was looking for, gives me a lot to think about. Thank you! 😁

If your main class is Rogue and the campaign goes long enough you might want to go as many as 5 levels into Bard. Maybe. That would get you a d8 Inspiration Die, Font of Inspiration, and 3rd level spells. But I wouldn’t go higher than that, and only if your Bard subclass is giving you interesting things to do with your inspiration die that fit the rest of your character concept. Otherwise Yurei’s right and it’s generally best to stop at 3 levels for your secondary class most of the time.

I've got a Bard/Rogue. Plan is to: Bard 1, Rogue 1, Bard 2, Rogue 2, Bard 3, Rogue 3, etc. etc till Bard 6/Rogue 6. If the character is still going I'm not sure where I'll go from there.

At Bard 3 I'm taking College of Swords and Rogue 3 I'm taking Swashbuckler.

Swashbuckler has the most synergy with Bard, since it overtly gives bonuses based on Charisma, but for a character who specializes in Sleight of Hand techniques I would actually recommend the Thief Subclass. As long as you have good DEX it's basically all you need for the subclass. It has bonuses that specifically make a character better at using objects, picking locks, etc., as well as making your character more acrobatic overall. It's main feature is being able to use an object as a bonus action, which drastically increases the value of things like Oil, caltrops, poisons, etc. that normally require a full action to utilize and are often less useful than just outright attacking someone... now you can do both!

For Bard I would recommend the College of Whispers. It's basically "Rogue Light", giving you a lot of abilities that work great for Rogues. The only downside is that it's kind of uh... Dark. It involves consuming the shadows of your opponents and imbuing fear on others. Just depends on the flavor you want for your character, but if you want a cheery, happy Bard type character it might feel out of place. One of the biggest bonuses is the "Psychic Blades" feature, which allows you to spend a Bardic Inspiration to deal additional psychic damage after landing a weapon attack.  This can help compensate for the fact that your Sneak Attack damage won't be as high as it would have been if you had gone full Rogue.

Quote from Mairondil >>

I've got a Bard/Rogue. Plan is to: Bard 1, Rogue 1, Bard 2, Rogue 2, Bard 3, Rogue 3, etc. etc till Bard 6/Rogue 6. If the character is still going I'm not sure where I'll go from there.

At Bard 3 I'm taking College of Swords and Rogue 3 I'm taking Swashbuckler.

You do you, but I want to point out this does not help your proficiency bonus at all, nor getting to your first ASI. 

Quote from Mairondil >>

I've got a Bard/Rogue. Plan is to: Bard 1, Rogue 1, Bard 2, Rogue 2, Bard 3, Rogue 3, etc. etc till Bard 6/Rogue 6. If the character is still going I'm not sure where I'll go from there.

At Bard 3 I'm taking College of Swords and Rogue 3 I'm taking Swashbuckler.

 I wouldn’t. I would go Rogue 1-4 then start working in the Bard. But I would stop at Bard 5, that 6th Bard level does nothing useful for you. As a Rogue your bread & butter is Sneak Attack, Extra Attack doesn’t really help you, and Countercharm is useless. Being able to potentially hit Slippery Mind and 8d6 sneak attack will help way more.

Personally I think Arcane Trickster is still the way to go. Yeah it uses your Intelligence but that won't really come into play if you pick a couple spells that don't use your casting modifier. Booming Blade and Find Familiar are great to have on a Rogue but Bards don't have access to them. 

I would definitely go Swashbuckler Rogue with your build and in terms of Bard you could also take College of Valor or Swords. Valor You gain the ability to wear medium armor and shield you could couple that with medium armor feat and gain the extra 1 AC with a dex of 16 or higher and wouldn't have disadvantage anymore while wearing medium armor in stealth checks, as well as Combat Inspiration to help boost your party members adding the inspiration dice to their damage rolls or their AC rolls if they are being attacked. With Swords you get either dueling if you plan to use something like a rapier giving you plus 2 damage or duel wielding. Blade Flourish really can add some flair to your character and you could use those abilities coupled with your magician background. 

I'm doing a Scout Rogue/Swords Bard in pbp here on this board.  Basically a mobitiliy build (with some overlapping features) plus the outdoorsyness.  Kinda becomes a sort of Errol Flynn Robinhood.