Red is one of the difficult colors to work with, it doesn't cover well and can have other problems.
Most likely all that is needed is another coat of paint. This time cut-in a wall at a time and then roll it it out. Try to get the roller as close as possible to the ceiling and trim, this will help minimize the picture frame effect. The paint can also be part of the problem. What brand did you use? What sheen?
Using a medium-grey primer would have helped. The manufacture should have a recommended primer color for dark or overly bright colors. In the future ask about a recommended primer before starting.