How to interview for positions while still having a job?

I've been with my current job for 7 months now, but after realizing there's no room for growth and the job is turning into something different than what was described when I accepted it, it's time for me to move on. Since I'm the only one in the office most of the time, it's easy to schedule and talk for phone interviews, but after two weeks of applying, I'm starting to have in person interviews scheduled. How do people get around going to in person interviews while still having a job? I feel like my employers will start to catch on if I start requesting to come in late or leave early for appointments multiple times a week. Any advice on how to go about this will be greatly appreciated. Thanks!

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