The answer depends on whether you want actual captions or just some text in the video. The difference that subtitles/captions are actual streams in the video container or seperate files altogether that can be turned on and off, whereas normal text in the video is part of the video stream, so it's burned in and can't be turned off (which also means the playback decive/software doesn't have to support the subtitle standard that is used, so there's benefits to both methods).
If you want actual captions, you have very limited formatting options. The way it works is the subtitle file/stream contains only the timecodes and corresponding text for all the subtitles you create and very limited formatting information as well as some metadata. The appearance of the captions (i.e. font family, font size and position) is controlled by the playback device/software. This way, the captions can be optimized for different displays and controlled by user settings (for example, people with bad eyesight can set a larger font size).
If you want to control the display aspects of your subtitles, you can create Open Captions that are burned into the video stream. Make sure to select 'Open Captions' as the standard when you create the captions in Premiere Pro:
If you create Open Captions this way, you have more control over the caption display:
As you can see in the screenshot above (taken in Premiere Pro 2018), you can control the font size, family and style, as you asked. If you create captions this way, you can still export the captions as an individual file (File → Export → Captions…), for example to the SubRip format (.srt). However, this will again only export the timecodes and text, not the display options set by you. If you want to have different font families, sizes et c. in your video, you will have to burn the subtitles in, which you can do if you create them as Open Captions.
Text in Video
If you want the most control over appearance, animation et c. of your text-in-video use Text / Graphics layers. Working with text used to be really tedious in earlier versions of Premiere, but with the new Graphics Workflow it's rather straightforward and elegant. Take a look at this page for an introduction. Keep in mind though that Graphics can't be exported as subtitle streams/files, so they will always be burned in the video stream.