I had a lot of freedom to set my own schedule, as long as I carried out my assignments. At the beginning of the summer, I met with my advisor and had some meetings in which he taught me how to use the database that he'd constructed for all his primary source research. After that, though, we mostly communicated by email with me emailing whenever I had results and him emailing whenever he had something else to add to my to-do list.
On a typical day, I'd get up and try to be either at the university library (tracking down books, microfilms, etc.) or at Panera Bread (coffee shops are the traditional grad student's office) by 10am or so. A lot of what I was doing was transcribing and indexing data from the thousands of digital photos he'd taken the last time he was at the archives. All that I did while hanging out at Panera (endless caffeine refills!) I'd get food and keep working while I ate, so I never really took an official lunch break, just worked until about 2-4pm and called it a day whenever my concentration started to fade.
When I had to pull secondary sources, cross-check printed materials, consult microfilmed materials, etc. then I'd go into the library instead. Those days I did have to take a lunch break, but usually I'd bring my own food or eat at the library cafe. I'd still put in ~4-6 hours of work and I kept track of it on timesheets. I'm not sure if those technically ended up going to the university or the people who provided the grant - I just turned them in to the department office every other Friday.
I tended not to work weekends - I saved those for my own research - but I could have worked weekends if I wanted. The key was to do ~20 hours a week, but other than that I had pretty much complete flexibility in managing my time. If I wanted to do two 10-hour workdays I could have.
At the end of the summer, we did do an official debriefing and report of what I'd accomplished, but I'd mostly given him everything already at that point. IIRC, it was a bit of a combination of general advising meeting and summer job report.
I do have friends who've interned in more office-y settings - those were generally for long-standing projects with actual paid staff and permanent offices as opposed to faculty-driven research projects - but other than coming in a little earlier and taking an actual lunch break, their experiences were similar to mine. A lot of the work is self-directed, since you're expected to have a good idea how to conduct research already when you're a grad student.