Is there a separate Companies object, and how do I find all the contacts at one company?

There is no separate Companies object. Company is a field on the contact, which keeps the CRM simple.

To find everyone at a company:

1. Open your contacts list.
2. Add a filter on the field Company.
3. Choose contains and type the company name.
4. Apply the filter.

Typing the company name in the search box searches all fields and can return unrelated results, so the filter on the Company field is more precise.