Trout have been kept in aquaria, and fellow students used to do so when I was at university. There's no real magic to keeping trout, assuming you have an aquarium big enough for fish easily able to top 45 cm under captive conditions. The main issues are temperature, oxygen, water quality, and diet. Starting with temperature, you'll need a chiller. That's non-negotiable unless you have a very cold basement. Water temperature needs to be kept below 14 C/57 F even in summer. Winter can be colder than this, though not freezing; 10 C/50 F is fine.
Besides cold water, these fish inhabit well-oxygenated streams and rivers (or the sea, if you're keeping migratory trout). Water turnover rate needs to be very high, and the aquarium will need a filter that maximises the mixing of air with water. Needless to say, such fish will also die at the first sniff of ammonia or nitrite. Oddly enough, nitrate isn't especially toxic to trout, though as ever, the lower the nitrate level, the better.
Feeding isn't a big deal because trout do well on trout pellets. These are widely sold as cheap catfish food in UK fish shops.
Realistically, home aquarists probably won't be keeping trout because of their size and demands. The so-called Hillstream Trouts, actually barbs, of the genus Barilius are infinitely better pets, and do extremely well in spacious subtropical (not tropical!) aquaria. Some species look very similar to trout indeed, and their behaviour is virtually identical.
Cheers, Neale