You can get to trust level 3 by…
This is time and activity based, unlike the previous levels. In the last 100 days…
- must have visited at least 50% of days
- must have replied to at least 30 different topics
- of topics created in the last 100 days, must have viewed 25% (currently ~250)
- of posts created in the last 100 days, must have read 25% (currently ~ 10000)
- must have received 300 likes*, and given 300 likes.
- must not have received more than 5 spam or offensive flags (with unique posts and unique users for each, confirmed by a moderator)
- must not have been suspended or silenced in the last 6 months
*These likes must be across a minimum number of different users (currently 75 users), across a minimum number of different days (currently 75 days). Likes cannot be from PMs.
Not 100% sure these are accurate, but this is the most recent and complete list I can find… I vaguely recall that something changed at some point, but whether this is the prior version or the new version of the requirements is a little foggy. 
Reading a lot and some chatting with people, that basically sums it up.