I would absolutelly vote against propriatary solutions like LastPass which might do what they say, which also could send all your passwords in plain text to the operator, you can't know because you can't look into the program and check what its doing, and so can nobody else either.
I suggest using free software (free like in liberty) where you or someone else can look at the code and find out what its doing. One of those is KeePassX https://www.keepassx.org/ like LastPass it is also available for all your devices like your Mac or Windows computer and also mobile phones, etc.
It has also a password generator build in, but what it doesn't is sync between devices, it just saves a incrypted database file on the current device. You can sync this file yourself with iCloud, DropBox, GoogleDrive etc. But if you don't want to rely on a commercial operator you could do it like me and use SyncThing https://syncthing.net/ which does basically what DropBox does which is syncing files between devices, but it is free software (as in libre) and doesn't need a central server like the others do, it just syncs the files between the devices themselves peer to peer.
This way you can trust the whole chain because everything is free software and checked by many people so id doesn't do evil. A nice byproduct is also that everything is free as in beer, so you don't need to pay any monthly fee or anything and you can be sure that this software will keep working even after the other companies stop providing those services.