Grain structure is an inherent part of shooting film. In general, lower ISO films have a tighter grain structure, allowing them to "look" smoother than higher ISO films (up to a point, if you're blowing up even an ISO 50 35mm frame to 24x36, you're going to see some grain).
Now, if you think that your image is overly grainy for the ISO that you have or the film type that you shot...that could very well be the case.
A proper exposure is needed along with proper development to minimize the grain size in any particular image. Color negative film (like your Colorplus film) handles overexposure very well - but it suffers from underexposure.
If the camera that you were using doesn't adjust exposure based on available light (many simple point and shoot cameras don't, or have shutter speed minimums that might still be too fast for the lighting conditions) then you are absolutely right in assuming that your film was too slow for your environment. These types of cameras are great for taking snaps - but with the limited ability to set exposure, you really do have to grab the right ISO for the lighting (100 for outdoor, 800 for indoor/dim, and 200 or 400 for level in between). Keep in mind that color negative film tolerates overexposure really, really well - so you want to err on this side.
As for the exposure on the print, keep in mind that the whole roll is being printed by a machine where a single setting was probably applied to the whole role. This means that, if your exposures were not consistent from shot to shot, then you'll have wide variation in your prints. You may want to take the image back to the lab with the negative and ask them to reprint, taking into account the single frame's exposure.