The longest running winter sports destination in all of Turkey, Uludağ is by far the most intriguing taste of elevation that you'll get when in Istanbul. This comes with a caveat, however. Due to its location on the opposite side of the Sea of Marmara (near the city of Bursa), this 2,543 m (8,343 ft) mountain is no easy skip and jump from the big city, with most transportation options still resulting in about five hours of travel. Still, when this is the closest option for skiing, snowboarding and quality hiking at elevation, you may have to make a few compromises.
Uludağ owns some heady historical context. Its original name was Mysian Olympos due to its place as the home of the gods, while it retained its religious mythos during medieval times as a place of Christian hermitage, earning it the name of Keşiş Dağı (Mountain of the Monks). But let's get to why you're here: the adventures. No matter the season (although October to April is your skiing window), you'll be getting great views of Bursa from Uludağ's broad plateaus. There are something like two dozen hotels operating on the mountain that will provide daily passes for the lifts, as well as a few vastly overpriced cafes and restaurants for those who aren't into skipping meals. A day on the slopes will run you about $40 USD, while it will be less than $15 to rent equipment. Hikers will want to head east of Oteller (where the hotels are) to the abandoned Vikram mine, where you can follow obvious but unmarked trails to the summit or to a collection of four glacial lakes. Both hikes will take about half a day there and back and can be managed in any season without climbing equipment.
As for getting there, you'll find that quickest access is via ferry from Istanbul to Mudanya, a dolmus into Bursa, and then the gondola line (Teleferik) from the Teferrüç neighborhood all the way up to Oteller. Then again, you can drive from Istanbul if you like; it's just a bit more roundabout.