We have two first round picks- number 6 and number 17. There are two rounds. You can only pick one player at a time.
The draft will be some time in June. About a week after the NBA Finals. Check NBA.com for dates.
In general, you want the highest pick possible(i.e. number 1). The higher the pick the higher the value. You can keep it or trade it. Yes, you can trade down if you like.
How important this year's draft is to the franchise? Well, the short answer is - you won't really know if you messed up right away. Every draft is important. I can tell you this, Danny Ainge is a good GM with a good head on his shoulders so I am confident.