What you really need is a dual-purpose drive, one that has high specific impulse for fuel efficient cruise (IE, ion rockets) but capable of a high thrust output for quick acceleration (something akin to the SABRE precooled turborocket). Maybe an ion rocket for cruise with a SABRE afterburner for escape velocity and rapid acceleration.
well cruise for a spacecraft is unpowered so no engines required for that. but it is quicker to accelerate fast then cruise than to use constant acceleration (more time spent near c, so you get more of the relativistic benefits of time/distance contraction).
for constant 1g acceleration the whole trip to Gliese 581 takes 6yr/22yr.
using the same fuel/payload ratio as above you can burn at 2g for 5ly, cruise for 10ly then decel at 2g for the last 5ly, in which case the trip to Gliese 581 takes 3.91yr/21yr.
point is that return trips to our nearest stars are completely feasible (and quick for the astronauts), and could be completed within the lifetime of an earth observer. and this is using inertial drives - no worm holes, subspace, gravity rails, FTL etc. just using developments of existing technology (ion drives, GRASERs etc.)
like the lunar trips, this is essentially just an engineering and resources challenge