The best thing about Galata is that you can spend two days here without getting into a vehicle. The neighbourhood is small, steep, and everything is close to everything else. This plan is built on that.
Day one, morning. Walk out of the door and downhill to Karaköy. The walk along the water takes twenty minutes, and seeing Istanbul this way first is a better opening than any museum. Have breakfast in Karaköy; menemen and strong tea are at their best before eleven.
Afternoon. Come back up towards the Galata Tower. If there is a queue, do not join it — you do not have to go up. The view that matters is in the streets around it: the old apartment houses on Serdar-ı Ekrem, the instrument makers on Galip Dede, and the small workshops in the lanes between.
Evening. Go up to İstiklal, but do not stay on the avenue itself; the places worth finding are on the side streets. After dinner walk towards Tünel, where the crowd thins out.
Day two. Take a ferry across. Karaköy to Kadıköy or Üsküdar is half an hour on the Bosphorus for a quarter of what a tourist boat costs. If you want the Historic Peninsula in the afternoon, cross the Galata Bridge on foot — five minutes, and the fishermen underneath it are a stop in themselves.
Late afternoon, if you would rather not walk back up the hill, the Tünel funicular does it in two minutes. It is the oldest underground line in the city and it still runs.
One warning: the neighbourhood really is steep. Bring comfortable shoes and plan the day downhill rather than up.