We will need nuclear propulsion for a lot of that, but the good news is that it has already been built and demonstrated back in the 70s.

edit: Rate limited so adding link to interview the reply mentions here: https://www.youtube.com/watch?v=ujfF_SnxvZQ

And NASA is working on it again! Isaacman talked about it in a recent interview with Everyday Astronaut.

I think the concept for the Alpha Centauri probes uses solar sails and externally fired lasers[1]. The exo-planet viewer concept also proposes solar sails.

1: https://en.wikipedia.org/wiki/2069_Alpha_Centauri_mission

2: https://en.wikipedia.org/wiki/Solar_gravitational_lens