For Pathfinder specifically,
As Eugene notes, Alchemical Power Components are your obvious first step - at +1 damage, for a one-time investment of under 50 GP (exact prices vary), you should always make sure to keep one on hand. 2-4 damage.
The Elemental Pupil trait gives you a +1 to all elemental damage with your chosen element, bringing it up to 3-5
The Havoc of the Society Trait adds another flat +1 damage (and Force, rather than Acid or Frost or Whatever damage), so we're now at 4-6.
Now, we still haven't chosen our class yet - and all of the above are valid on any level one character. If we want to maximize our Cantrip damage right out of the gate, Level 1 should be as a Sorcerer with the Crossblooded Archetype. That'll allow us to select two Bloodlines, and get the level one Arcana from both of them. Which is great, because the Orc Bloodline and the Draconic Bloodline offer near-identical bonuses that stack - each will add +1 damage per die (just one for our cantrip) of the spell, for +2 damage. We're now at 6-8 damage at level 1 - a respectable spammable attack!
However, this also marks the point of No Return, because now we've chosen our Cantrip, as the Draconic bloodline restricts it's bonus to damage of a specific element. I'm going to suggest Ray of Frost, though if you're not happy about doing cold damage, you have some options - see below.
Because Ray of Frost is a ray spell, and therefore is 'weapon-like', it should work with Point Blank Shot for an additional +1 damage if fired from within 30 feet. 7-9.
At level 2, we're going to start taking levels of Wizard rather than Sorcerer. 18 of them in fact, eventually, because with the Evoker school specialization, we can add half our wizard level to the damage of our Ray. (If Cold damage isn't really your thing, you can take the Admixture sub-specialization and use whatever element you feel like.) At level 2, that puts us at 8-10, but by level 19, we'll be at 16-18.
As an alternative, we could stick with Sorcerer levels - but be stuck playing a Dwarf - and take their great Favored Class Bonus to add +1 damage to Acid and Earth spells per Sorcerer level. Comparable results to the Wizard option above, but you're swimming uphill against stat penalties, and Acid Splash isn't quite as versatile a spell as Ray of Frost. You can accomplish something similar as a Half-Orc with 1 level of Admixture wizard and 18 levels of Sorcerer, turning Ray of Frost into a Fire spell. (In fact, the Half-Orc comes out one damage ahead I think.)
Now, as we level up, we can pick up a few more feats that will work with Ray of Frost, since, for all intents and purposes, it's a weapon, and feats that work on weapon damage, work for rays.
Arcane Strike is the big one, as it will scale with you, adding +1 to damage at level 1, and for every 5 caster levels, adding another +1 damage.
If you stick with Ray of Frost and Cold Damage, you can do Sorc 1 / Wiz 18, and one level of Witch with the Winter Witch archetype, taking the Frozen Caress hex at level 1, to add 1d4 damage to your Ray of Frost. Bonus, since it's an extra die of damage, it should qualify for a boost from your Bloodlines, adding another +2, or +3-6 damage net. Keep in mind however, that this option is incompatible with Arcane Strike.
There are a few other roads you can go down at this point, to suit your temperament and desired build - just remember, any feat or effect that increases 'weapon damage' will work on a Ray spell, and that's the key here. Flag-Bearer is another notables one that provides a flat +1 to Hit and Damage. (Incidentally, if you can find a way to qualify for Weapon Specialization (Ray) without sinking too much time into Fighter levels, well, that's +2 damage. Similarly, dipping Ranger for Favored Enemy will boost your Cantrip.) You've got a lot of options, but at this point, at a minimum, it's always going to at least be competitive with a mundane weapon making a mundane hit. And it hits Touch AC for good measure.
(Of course you could always go the easy way out, ignore almost all of this advice, and just put a bunch of levels into Rogue. You can Sneak Attack with a Ray spell, and thus, just use Acid Splash against a flat-footed opponent from within 30 feet and do many d6's of damage. But that's kind of cheating. :P)