If you're looking to cross the mighty Bosphorus to the Asian side, the greater district of Kadıköy is undoubtedly where you'll end up. Although Kadıköy itself has quite a lot to do, the subsidiary neighborhood of Caddebostan is the best concentration of drinking, dining, and entertainment in the whole eastern half of Istanbul.
Many visitors will make their way to riverside promenade and the adjacent beach when the weather permits, and although there are some quality cafes near the waterfront, you'll want to follow the social buzz a good few blocks inland to uber-popular Bağdat Caddesi. Ironically, this street may have the most European feel in the city despite the fact that it is located on the Asian side, with its broad, tree-lined sidewalks, cafes, steakhouses, cinemas, bars, and even a couple of nightclubs. It's a bit more pricey here, but the selection is extensive and the lengthy boulevard just begs to be strolled.
Then again, if you're looking for possibly the greatest collection of nightlife establishments in the area, then look no further than the appropriately named barlar sokağı (Turkish for "bar street"). Officially Iskele Street, the twenty or so bars, pubs, and clubs found here shout out Western names (Sheriff, Alcatraz, Barcode) with a neon fervor. Think shot specials and dance parties late into the night and you've got this place nailed down. If a night on the town is your jam, then this incredibly lively handful of blocks from Bağdat to the river is a no-brainer.