Solu and Khumbu are in Solukhumbu district in Sagarmatha
zone.
The fastest and simplest way to get to this area is by
air. To reach the Solu region, you can take a half hour
flight from Kathmandu to Phaplu and then a 20-minute walk
to Salleri - the district headquarters of Solukhumbu.
To reach Khumbu there are daily flights from Kathmandu
to Lukla if you have limited time. However, proper acclimatization
with rest days and a slow but steady pace is strongly
advised. 
To reach Solu region from Khumbu and vice versa, one
has the option of trekking for about 2 days via Salleri-Tapting-Kerung
and via Nunthala-Kharikhola-Chaurikharka. One can also
fly which is rather cumbersome as one would need to fly
out of Phaplu in Solu to Kathmandu, and then catch a flight
back to Lukla in Khumbu.