You can see in my profile that I've visited the site 908 consecutive times out of 1097 days that I've visited it, so I've been visiting the site at least once per day for more than 2 years.
Having said that, I am a bit chaotic and irregular, so I do not visit Artificial Intelligence Stack Exchange always at the same time or use the same approach. I can visit the site and contribute to it (e.g. by answering questions or editing posts) in the morning or late at night. If I feel bored, I may visit the site more frequently.
One of the first things that I do when I visit the site is to take a look at the latest questions. If I have some time or feel bored, I try to read a few of them, maybe leave a comment asking for clarifications (if that's necessary), edit the post if it contains mistakes or its clarity and readability can be improved, or maybe upvote/downvote it (depending on the quality). If I know the answer to the question and it can be written in a short time (and I have time to write it), I may decide to write an answer immediately, but sometimes I leave some time for others to answer it, so that it's not always me that answers it, or I may keep a tab opened for a few days, until I have the time to answer it (sometimes, those tabs are closed to reduce the noise or because I lost interest in those questions). Sometimes, I do not remember or know all the details needed to answer an interesting question that I want to answer, so I may leave a tab opened with the question, so that I can answer it later, after having done a little bit of research (e.g. take a look at a reference book or paper) that contains the details that I forgot or don't know.
Occasionally, I may also read posts that have popped up in the "Top Questions" section (i.e. the section that appears when you visit our site), which may also contain old posts "bumped by the community user". Moreover, sometimes, I also read posts that are connected to a post that I'm already reading (right section). For example, if I see a post with a weird or unclear title, I may read it, in order to try to improve the title. I've done this many times in the past. That's also why you see many posts edited by me.
As a moderator, I also have other responsibilities. In particular, recently, I've been trying to verify that the posts closed by other community members have been correctly closed. See this. Occasionally, I also look at the queues, but I leave this job for other community members or moderators, as I already spent a lot of time reading the latest questions/posts. There are also other tasks that I may need to do as a moderator, but I don't want to dwell on that here, as the question doesn't seem to be about what a moderator may be doing in the background (and I couldn't and don't want to share everything anyway).
Similar to what Neil does, I also have a few private chat rooms where, thanks to the RSS feeds, I can regularly read about posts tagged with topics that I may be interested in. Often, I forget about these chat rooms, as I don't have much time to read everything.