Building a gnome illusionist

I’m looking to get some ideas on building a gnome illusionist. This character will be using the free archetype variant and can use all (including uncommon and rare) official character options. I expect them to get to 20th level. I am open to all options, including a universalist acting as an illusionist or the other extreme of runelord specializing in illusion. Additionally, I’m looking at some synergistic archetypes like spell trickster or something similar. Has anyone had any experiences using an illusionist and had any combinations that worked well and others that didn’t?

Illusionists are great in this edition. Illusions explicitly need to be interacted with before they offer a save meaning even if they critically succeed, they've already wasted an action to do so.

Gnomes, on the other hand, are pretty mediocre. Their ability boosts are bad for a wizard since you need to boost dex/con/int/wis at every opportunity. You wind up with a wasted cha boost and a reflex and AC penalty from 1-9 and 15-19.

But, gnome illusionist is a classic so I assume you're not willing to budge much beyond what you yourself stated. There's not much to it. Be a spell blending illusion specialist, grab an illusion staff (for illusory object mostly) and you're done. Wizard feats are mostly garbage so you should spend most of them on archetype feats.

For archetypes, you have a few options. Arcane witch will give you more slots for illusions, a useful focus spell in life boost and a familiar (YMMV). Alchemist is good for some extra utility potions/elixers. Psychic offers a good reaction with amp guidance or pf1's dimensional slide with amp warp step. Gunslinger offers a good reaction with fake out.

I would start with:

That'll give you a worthwhile focus spell, a few different reactions including one you can use every turn, an initiative bonus, master perception, utility alchemy items and the usual focus pool and sustain feats.

Convincing illusions is skipped because you aren't convincing anyone with 12 cha. That feat is better on a half-elf cha caster using multitalented at 9 to get wizard dedication without needing int investment.

If only interested in the illusions (not the sub-class), consider being a Sorcerer w/ illusions. Occult or Arcane work best, and I'd tailor the option so there's juxtaposition with the other casters (Occult is opposite Primal, and Arcane is opposite Divine).
Bard also works, especially if the party leans toward martial PCs, though you'll have fewer illusions to cast (yet you'll be tougher too).

If going Wizard and expecting to be active in civilized settings, consider Conceal Spell & Silent Spell. If only dungeon delving, don't. Raving Dork has had immense fun with those feats (though funnily enough he had his Sorcerer take Wizard MCD to get them, meaning he even spent higher level slots.) These sync well with Invisibility Cloak & illusions for city shenanigans.

I was about to suggest going Runelord while you have the opportunity. Lean into it, right? But I have some doubts. It does give strong abilities for little investment, but its feats are mediocre. Since some are skill feats (albeit bad), you can exit swiftly into a new Dedication so weigh how much you like your Focus Spells.* And you'd want to make sure a party member has access to key conjuration & transmutation spells, i.e. Haste, though a party could build around the lack.

*Most casters should have at least one Focus Spell they like! If necessary, you could always pick up excellent Focus Spells from another class, though not many sync well with Int/Arcane. IMO Blessed One would give a worthwhile Focus Spell and open up some interesting options (mostly since you're going Free Archetype AND into the highest levels where Blessed Denial would shine when needed).

Since you're stuck with a 12 Cha, consider getting 14 so you can gain entry into an interesting Dedication. Many on the boards here love One for All from the Swashbuckler for Aid at range. Bard MCD for some more Cantrips and access to Compositions.
And depending on the campaign, all of those languages from Int might prove useful enough that you want social skills too (which kind of suggests going Rogue MCD to pick up skill feats & since you're there some defensive feats like Mobility).

Gesalt is correct about gnomes being under the top of the power curve re: wizardry, but there's ample space for success unless running the fiercest of campaigns. 18 Int is 18 Int, which covers 90% of your PC's value. IMO you're just trading unhelpful Str for barely helpful Cha, and getting some interesting Ancestry feats along the way.
If your campaign is running on a theme, Gnome Obsession is a good investment, i.e. pirates and Lore (sailing), though the magical tricks are the neatest IMO, like Project Persona.

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After saying all that, were it a typical campaign I'd switch classes. But given yours I'd go with a Wizard/Runelord/Blessed One. How much I leaned into Cha & social skills (or Conceal/Silent) would depend on the campaign plus how well other party members cover those bases. Though I wouldn't expect to act as primary face, there's often room to help there (as long as you also get your Dex/Con/Wis to 18!)

Gnome Summoner with Fey Eidolon is the illusion path that I plan on going for the right AP.

With Free Archetype, most likely just go Sorcerer for more slots...

In EC, I went with Goblin (Lich) Wizard (Universalist).
He had all the Illusion Feats, Stealth Feats, and the Cha to pull it all off.
He primarily used Illusions early in career, but since about 5th gravitated more towards Summons, which were really fun.
We are 20th now and in Round 4 of The Boss fight....