The Summer Palace is the Qing court's idea of escaping the city, and even with the modern capital spread out around it, the idea still works. The core is Kunming Lake, a large artificial basin, with Longevity Hill rising from its northern shore and covered in halls, pavilions and the tall Tower of Buddhist Incense. The Qianlong Emperor shaped the garden in the mid eighteenth century, reportedly to celebrate his mother's birthday, and modelled parts of it on the scenery around West Lake in Hangzhou. British and French troops burned it during the Second Opium War in 1860, and the Empress Dowager Cixi rebuilt it later in the century, which is where the famous story comes from that she paid for it with money meant for the navy. The marble boat moored at the lakeshore, which cannot sail, has become the punchline of that tale.
The Long Corridor running along the shore is the thing to walk, painted with thousands of scenes from legends and novels on its beams. Go early, ideally right at opening, because by late morning tour groups fill it shoulder to shoulder. Winter brings a sharp, clean light and sometimes ice on the lake; spring has blossom and the sandy wind from the north; summer is hot, humid and crowded.
As for the riding, be realistic. The palace sits in Haidian district, northwest of the centre, inside the urban sprawl, and Beijing places strict limits on motorcycles, especially those not registered in the city and especially inside the inner ring roads. Check the current rules before you plan a route, and expect heavy traffic, electric scooters on every side and very little in the way of enjoyable corners until you head out toward the Western Hills. Once you do, the roads climbing toward Fragrant Hills and beyond into Mentougou get properly interesting, with switchbacks and temples tucked into the valleys.
Fuel, food and every other service are everywhere around the palace; this is a city of millions. There are noodle and dumpling places within a short walk of the gates, and the area near Peking University and Tsinghua to the east has cheap student food. Park legally, keep your helmet with you and give yourself half a day inside: the grounds are much bigger than they look on a map.