The most important job in brewing is cleaning and sanitizing your equipment. The level of cleaning you need to perform each time you brew is specific to your setup, your schedule, your equipment and how messy the brew was. That being said, here's my usual routine:
Cleaned and sanitized the day before brew day: Anything that will touch the wort or beer AFTER it's been brewed (fermenters, spoons, hydrometers and dip tubes, auto siphon, strainers, hops bags, blow off tubes, airlocks, 02 wand, wort chiller, etc.
Cleaned but not usually sanitized ON BREW DAY: Mash tun, boil kettle, thermometers, mash paddle, BIAB bag.
Cleaned AFTER BREWING on brew day: boil kettle, mash tun, BIAB bag, auto siphon, thermometers, O2 wand, hops bags, strainers, spoons/stirrers, mash paddle and wort chiller. Not sanitized, just cleaned of all grains, trub, gunk and malt residue.
Cleaned and sanitized BEFORE Bottling/kegging: Bottles, kegs, caps, capper, priming sugar (simple syrup), bottling bucket, spigot, filling wand, auto siphon.
Cleaned but not sanitized AFTER Bottling/Kegging: Fermenter, auto siphon, adjunct ingredient bags (usually hops bags), blow off equipment, airlock, bottling bucket, filling wand, spigot, etc..
As far as your questions, it's ok to just rinse out your hydrometer tube and hydrometer if you don't dump your beer back into the fermenter. Anything going into the beer should be cleaned and sanitized though. So, if you're using a hydrometer tube, an auto siphon or a wine thief (or big turkey baster) to take samples of the beer for the hydrometer, make sure that's sanitized.
I prefer to clean up my equipment right after brewing for a few reasons: Clean equipment and brewing environment reduces the possibility for infections, it prevents molds and fungi, it reduces rust and oxidation and it doesn't take but about 20 minutes of my time after the brew is done. I don't sanitize anything after brew day, just remove the major residue and gunk from my equipment so that it's clean. For a few pieces of the equipment like my O2 wand, this is vital for it's long term usability.