I only have one budgie, but hopefully I can offer something useful!
I've had my parakeet for about... 7 or 8 months now? and he lives by himself in a big flight cage. Originally, I wanted to get a second budgie but then I got super stressed out about quarantine and bird illnesses (Long story there) and decided against it. He seems to be very happy, and he has lots to do (foraging toys, natural perches with bark to peel off, shedding toys, etc) whenever I'm not with him.
Now my situation is a little bit different than yours. I'm homeschooled, so I'm home almost all the time unless I'm off running errands with my dad or something. Since you're in school, I would think that you current budgie would probably like to have a friend while you're gone during the day. However, keep in mind that you'd have to quarantine the new one (in a separate cage, in a separate room (or preferably a different building. although that's not feasible for a lot of people, including myself) for at least 30 days, but ideally 90 days, with a vet visit at the end of the quarantine period to make sure the new bird is healthy. Then you'd probably need to go through a period of introducing the birds to one another, and then if they get along, introducing the new bird to the cage he'll be living in with your other budgie.
But in general, yes, budgies are birds who prefer to live in groups. In the wild, they live in huge groups, of hundreds or thousands. They're very social, and they love interacting with their human flock members (at least mine does!) I'm sure that if you got another budgie, they'd probably be very excited to be with another bird of their kind.
And on the training question, I don't see why you couldn't have them live together when you're training the second one. I haven't done a whole lot of training with my budgie, but I would think if anything the second budgie might pick up some things from the first budgie when you're working with him!