I'm printing a large poster through a full color inkjet plotter.
What is the best format to export a design from illustrator for print, that will result in the best quality and lowest file size?
PDF format is the most versatile and flexible format. It can be used for both high-end printing (litho) low resolution web use and in between depending on the settings.
In illustrator save dialogue boxes you can set the image compression. I would suggest no higher than 300dpi. You can also set the compression level (generally use "high" but not "very high" jpg compression.
One important tip: use "save as" and make a new file with "preserve illustrator editing capability" switched off otherwise your file will be much larger than necessary. This new file will not be editable but will be optimized for file size and print quality.
"Best" isn't really a definitive term. And "lowest file size" is often in direct competition to quality.
If you want optimum quality printing to an inkjet machine, send RGB data at 300ppi or better. File size (kb) should not be a factor you worry about.
Depending upon what you are outputting from there may be other considerations. For example if you intend to print from Adobe InDesign or Adobe Illustrator, you'll want a Postscript Level 3 RIP somewhere. Some inkjets have a RIP, some don't. If you don't, then export/save as a PDF and print the PDF. Acrobat is a software RIP.