Could someone recommend a short cruise either one way or 2 ways from the European side to the Asia side?

There's so many online that I'm getting confused...

My recollection is that there is a standard ferry that goes to the Black Sea mouth of the Bosphorus. The trip would take the better part of the day, especially if you decide to hike up to the fortifications overlooking the area.

There are tour boats leaving the Old City constantly -- if you are anywhere near the Galata Bridge, the touts will be all over you trying to get you on their boat. Different boats go different places so make sure you know what you're buying. The most scenic Bosphorus tours go north (away from the mouth of the Bosphorus) past Ortakoy to the very scenic Rumeli Fortress, and then back. GO EARLY IN THE MORNING. 90% of the interesting stuff is on the European side, so in the morning the sun is behind you, giving you the ideal lighting to look at everything in detail. By noon, the hot sun is blazing overhead and you can't see the detail. And in the afternoon you'll just be squinting into the sun and you won't see anything, If the boat makes stops en route, you could consider leaving the tour on the return trip either at the Rumeli Fortress or the scenic district of Ortakoy, having a look around, and then get a taxi back into the Old City via the scenic road that runs right along the Bosphorus shore.

Heading the opposite direction (from the Old City south to the mouth of the Bosphorus and out into the Sea of Maramara) boats are leaving constantly for the four Princes Islands, which make an excellent day trip (I went to Buyukada and loved it, but I hear that all of them are nice). These are lightly populated islands with stately 19th-century homes and only horses and carriages for transport, a trip back to the 19th century. Buyukada also has a very high overlook that is a tough climb, but well worth it.