Advantages of large firms:

1. More people = more networking opportunities
2. More people = higher chance of meeting people you actually like
3. Likely to be paid a higher salary
4. Will be more resistant to market fluctuations and difficulties (you’re less likely to get fired in a recession)
5. Ability to move to other offices around the world.

Advantages of small firms:

1. A great sense of impact. You are more likely to see the results of the work that you do, making your career more rewarding.
2. The company is a lot more personable. You will likely get to know everyone in the company, giving it more of a “family feel”
3. Potentially a faster route to more senior positions, due to a lack of competition and strict company rules.
4. Likely to be specialised to a certain niche. This means you will become more qualified more quickly. (Much better if you already know what you want to work on e.g. healthcare)

This article has a good explanation of the factors you should consider when picking a firm: Why should I do consulting? – Consulting Hack

Overall I’d say definitely start at a larger firm, and then move to a smaller one when you’ve found what industry you want to specialise in.