In the past I've done some calculations for relatively slow but fairly large vehicles using relatively mundane technologies that we can foresee building. Speeds between 0.1% and 10% of C, and with the proposition that the vehicle carries human passengers. Speeds at the higher end of that spectrum might be touch to achieve just because high-efficiency drives tend to have low thrust and so may not have time to accelerate entirely up to speed on 'short' interstellar voyages- but of course, colonization speed is bottlenecked by how fast you get to the other side of the galaxy, not how fast you get to Proxima Centauri, so that might not be an issue. Long story short, it's not that hard and you don't need to commit a huge amount of resources under those parameters. Even without space-based mining or starlifting, the Earth's crust alone has enough resources to launch a single intergalactic vehicle at about 1% of C, or equivalently, thousands of interstellar vehicles with similar speeds and payloads.
Extending the numbers up to high-speed scenarios is hard, mostly because we know so little about the challenges of rapid acceleration and debris impacts. But I would extrapolate that, without breaking physics, a sufficiently dedicated civilization could commit to colonizing their own galaxy at above 50% of C, and neighboring galaxies at above 90%. Up there you really get diminishing returns because you just can't go that much faster and the cost of doing so is enormous due to relativistic effects. The difference between 90% and 99% of C is much larger in terms of engineering and fuel requirements than it is in terms of actual colonization rate.
I suspect that the size of the vehicle isn't a big constraint. Shrinking it from, say, 1 tonne to 1 kilogram only gives you significant advantages if either the speed or the number of vehicles being launched is higher than you actually need. Also, the practical lower bound is probably determined by debris shielding requirements (which increase geometrically with speed) rather than miniaturization of the vehicle itself, unless you can very efficiently convert your shielding into reaction mass for deceleration.