First, if you are editing raw in Lightroom you are already using LR's internal color space for edits. Unless you export to Photoshop, you don't have a choice of editing spaces...
Second, unless you DO calibrate your display you'll have no idea what your images really look like. So, yes, get a calibration/profiling solution.
Third, yes, soft proofing allows you to predict what the image will look like in print. And currently that means Photoshop. As for the color space, you'll need to decide but I use ProPhoto RGB in 16 bits.
4th, if using LR to print, you either set the Print module to have the Printer Manage Color (not usually optimal) of LR managing color. If you have LR manage, you set the profile for the printer/paper you are using.
You still have a long way to go before you understand this stuff...maybe some reading or tutorials would help. The current Camera to Print & Screen here on LuLa would be a good place to start...(course, I might be biased since I was involved with making them).