June is the rainy season and the road conditions will be very bad. Generally, you try to visit Zero point early in the morning as weather will get bad in the afternoon. From Lachung it takes 2 to 2.5 hours if you don't stop for long in between. Yumthang valley is on the way and close to Lachung, so you can cover them in a single morning. After lunch you can move to Gangtok or any other place or opt to stay at Lachung as well.