I know that in Chinese we can say "竞选” for campaigning （as in a verb). But, if I wanted to say "campaign" as in "Obama's Campaign" how would I say that? I've seen 竞选活动 in dictionaries。 But， would people say 奥巴马的“竞选活动“？
Yes, you can say 奥巴马的竞选活动.
竞选活动 is referred to the activities in lie to some competitions like elections.
this is an example from dict.cn
- 在明年的重选中她参加"竞选"。 She comes up for re-election next year.
- 他在"总统竞选"中曾以改革者的姿态出现 He have project himself as a reformer in the presidential campaign.
- 有人说布什因选举大会开得不好以及"竞选"方法不佳而受害。可是即使两者都做得较好，结果也将是一样。我想根本问题也不在于经济。 Some will say Mr. Bush was done in by a bad convention and a bad campaign. But if both had gone better the outcome would have been the same. And I don't think it all came down to the economy, either.
- 他将参加"总统竞选"。 He'll run for the Presidency.