Best used car under $5k for a first car?

For a budget under $5,000, you're looking at older used cars, but the data here doesn't support a reliable pick. The Toyota Corolla and Honda Civic both have typical prices far above your limit, median around $20,000 and $21,000 respectively, so any Corolla or Civic you find for $5k will be very old or high-mileage, and the owner reports show serious problems in those older years.

The Honda Civic has recurrent engine issues in older generations: cracked engine blocks on 2006-2009 1.8L engines and head gasket failures on 2001-2005 models. Rust is also a common problem on older Civics. These are major repairs that would blow your budget. The Mazda 3 has no price or issue data provided, so I can't evaluate it.

Given your situation, none of these vehicles are a safe bet. You'd be better off looking for a reliable, low-cost car outside these options, like a used Toyota Yaris or a well-maintained older sedan from a brand with cheaper parts. Focus on a vehicle with a known history of minimal repairs in the $3,000-$5,000 range, and get a pre-purchase inspection from a mechanic before buying.