Starting March 16, LastPass's Free plan password manager will be essentially unusable; see here for more details. Only users that don't have a computer (i.e., have only phones and/or tablets) will find this new limitation of LastPass Free acceptable; everyone else will need to move to something else.
I have recommended LastPass Free for years but that's not going to work now. If you're looking for a free replacement for LastPass Free, and you care about security and privacy, Bitwarden Free looks like the best bet. If you search you'll find a huge number of glowing reviews on the web. It's not too hard to export your vault from LastPass and import it to Bitwarden.
However, if you're happy with LastPass, consider staying with it and upgrading to LastPass Premium, for USD $36 a year. In addition to more powerful sharing, 1 GB of file storage (instead of 50 MB), more 2FA options, and tech support, you'll get the feature that I recommend everyone make use of: Emergency Access.
Bitwarden Premium, USD $10/year, also has an Emergency Access feature.
It doesn't matter which you use -- LastPass or Bitwarden or something else -- but it's essential that you use a password manager, and use it properly. See my blog post on this.
---Update 2023-01: LastPass had a big data breach in 2022, and appears to no longer be a good choice.
Update 2021-03-16: If you're part of a family or have a small group of like-minded friends, the LastPass Families or Bitwarden Families plans can save you a lot of money.