You should bleach it first, but be careful, because your roots will end up much lighter than the dyed parts if you do this... So I suggest either putting bleach on the dyed parts for longer, or just dying the red over what you currently have. The red won't be as bright if you do this, but it'll probably have a more even effect than bleaching first, as I did this and my roots ended up much brighter than the rest...
In my experience as well, hairdressers often don't know what to do with hair like this themselves - I've had worse dye jobs from them than I've done myself before... So if you're going to splash your cash on one, make sure they know exactly what you want and that they know what they're doing.