How many journals can one follow at the same time?
Use a content manager and subscribe to the journals' RSS feeds.

I currently follow ~30 journals and a few blogs (this number changes dynamically as I remove some journals and include others) using feedly, which is a RSS reader that is very user-friendly and has apps for both Android and iOS. The software automatically fetches new content from the journals' latest issues and makes them available to me in real time. My daily feed is usually comprised of ~50 new papers every day. I devote 15-30 minutes of every morning to going through it--except on some weekdays when journals like Nature and Science release an entire issue. Going through the feed includes reading all titles, identifying potentially relevant papers, reading their abstract and saving them to my Mendeley library. Some people keep track of new papers through Mendeley alone, though I prefer feedly.