Ordered from easiest to reach — out of the city into real Shandong
Closest to furthest — we say plainly which you can do on the metro yourself, which need a high-speed train, which are a half day and which are a full day or an overnight, and which suits a single free day.
1
If you only leave the city once and don't want to travel far, Mount Lao is the answer. This is China's tallest and largest coastal Taoist mountain, about 30–40 km east of the centre. The favourite spot is Taiqing Palace (太清宫), an ancient Taoist temple right by the water, tucked beneath granite cliffs with the open sea in front of it. Around it are Mingxia Cave, Longtan Waterfall and ancient trees centuries old — the one place where you get mountain, temple and sea on a single trip.
What makes it stand out is that it's the easiest of the three to do yourself: it's close, and Metro Line 11 runs along the coast to the Laoshan side with lovely views the whole way. The catch is that the park is vast, and inside you ride shuttle buses between the sights in stages, so if you'd rather not change buses several times, a half- or full-day tour with transport is more comfortable.
2
If you want a trip you'll remember with a story behind it, this is it. Penglai Pavilion (蓬莱阁) is a cluster of ancient halls perched on a sea cliff in Penglai, in the Yantai area. The Chinese count it among the country's four great pavilions, and it's the legendary home of the Eight Immortals crossing the sea. It's also famous for the sea mirage that genuinely appears on the right days — which is how it earned its 'fairyland' name. Stand on the pavilion and the Bohai Sea opens out as far as you can see.
Yantai itself is a pleasant seaside city with an old colonial quarter and wineries (Yantai is one of China's best-known wine regions). A high-speed train runs from Qingdao straight to Penglai station in about 1.5–2.5 hours depending on the service. To be straight with you, it works as a day trip, but a full one — you'll want an early start and a clear plan for the local buses; if you have the time, an overnight in Yantai is well worth it.
3
If you love an old Chinese city with springs and a lake at its heart, go to Jinan (济南), the capital of Shandong. It's nicknamed the city of springs because hundreds of natural springs bubble up across the old centre. The headline is Baotu Spring (趵突泉), rated the city's number-one spring, where clear water gushes up in constant fountains — long a favourite of emperors — and Daming Lake (大明湖), a lake in the middle of the city fed by those same springs, perfect for a willow-lined stroll.
High-speed trains from Qingdao or Qingdao North reach Jinan in about 1.5–2.5 hours, with dozens of departures a day from morning to night and the fastest at about 1 hour 40 minutes. To be honest, it works as a day trip but a tiring one, since the sights are spread out; if you want a relaxed pace and a taste of the old city at night, stay an overnight.