If you haven't gotten some cleaning kit yet, I'd recommend the following basics as something everyone should have for DSLRs: a decent microfiber or lens cloth, a bulb-type sensor blower, and possibly a sensor brush or wet swabs. Not much more needed than that. Remember when cleaning lens glass, to make sure you don't have any solid particulates on the lens before you start rubbing it down - blow off any minor dust/dirt/etc, then gently wipe a clean cloth across the lens with one side, flip the cloth or use a different patch, then finish rubbing and cleaning as needed. When cleaning dust off the sensor (you'll know when you start seeing little dust spot in blue skies), a bulb blower will often handle 75-80% of the jobs...for those few times you have more stubborn dust, the sensor brush or last resort wet swabs can finish the job. With the camera body, just remember to occasionally wipe it down with a nice microfiber cloth - keep salt off the body, clean of dust occasionally, and be careful with any forms of lotions or moisturizers around the camera, as many of these left on your fingers even in discreet amounts can still result in losing much of the writing on your camera body!
As for portrait lenses, indeed primes are usually best in overall sharpness, and those with a decently fast aperture will be best for DOF control and shallow depth of field effects. One of the better ones you can pick up cheap is the Tamron 90mm F2.8 - a well respected portrait and macro lens that has just about the right focal length for portrait work, good shallow depth of field...and cheap too (usually can be found new for under $200 US, or even down to $100 used.
Sony DSLR-A580 / Sony 18-250mm / Minolta 50mm F1.7 / Sigma 30mm F1.4 / Tamron 10-24mm / Tamron 150-600mm / Tamron 90mm F2.8 macro / Minolta 300mm F4 APO
Sony A6000 / 18-55mm F3.5-5.6 / 55-210mm F4-6.3 / 10-18mm F4 / 35mm F1.8 / 16mm F2.8 / via manual adapter, lots of Pentax K mount, Konica K/AR mount, and Leica M mount manual lenses