Religions would face a huge obstacle in bridging their creation myths, more people would become atheist and non-religious.
Knowing something is possible is a hugely motivating factor. The department of defense and every international analog would want bigger and bolder weapons, money would pour into science. We'd fund astronomy and space exploration orders of magnitude more. There would be a renewed push for humans on Mars and beyond.
We'd want to know if alien life was biological or an AGI. We'd push harder on our own advances, but also become more aware of how this acceleration affects us.
We'd keep looking for life in our backyard. We'd be worried about it. Popular science and science fiction would communicate that risk and would lead to new cultural overtones. Zombies and superhero movies get replaced with alien invasion films.
I think we'd accelerate everything.
Edit: my post is proving controversial because of my statement on religion, which totally flies over my main point. With regard to religion: (1) I haven't made any value judgments, (2) the impact of science, technology, and modern culture has deceased religiously in the West measurably. I should think incontrovertible proof of alien life would only contribute to this trend.