How to insert values into a table from two different tables?
You would write the query like this

insert into classroom (date, teacher_id, student_id)
select '2014-07-08', t.id, s.id
from teachers t,students s
where t.teacher_name = 'david'
and s.student_name = 'sam';


Be careful. This is a Cartesian product. Another way to approach this is

select teacher_id into @tid from teachers where teacher_name = 'david';
select student_id into @sid from students where student_name = 'sam';
insert into classroom (date, teacher_id, student_id) values ('2014-07-08',@tid,@sid);