In manuscripts, should I indent the first line of a paragraph or can I just skip a line and not indent the first line? Indenting looks so old fashioned.

The age of the Internet has certainly changed the way writing appears visually. On-line, most writing is blocked, where the first lines of new paragraphs aren't indented, but are flush to the left like all the other text. White space indicates the movement to a new paragraph. This might be why indenting seems old fashioned to you. We don't often see it in the newer technology of the Internet. Much of the motivation behind this is visual. On-line, an article can unfurl on a single long page, so the space isn't as limited as it is in printed matter. In a book, too much white space can look awkward.

Manuscript format is to indent the first line of each new paragraph (and to double space). Don't skip lines when doing this, unless you're indicating a larger break in the story. This is the industry standard, so editors and agents are expecting it. And following the standard communicates to the reader that you've taken the time to learn this format, which is one way to convey your professionalism.