I have a fairly dated site that uses Trebuchet MS font in Latin and Cyrillic scripts. If I understand correctly, once (a decade ago) this font was available on like 95-99% of computers used for web browsing and it was ok to use it - nowadays myriads of devices (namely, Android, iOS, Linux-based devices) don't have it. In theory, it could be downloaded from Microsoft website and installed on per-computer basis, but it's not possible to redistribute it, i.e. make a web font from it.
I want to update that site and use a downloadable web font - preferably a free one. Can anyone suggest a good alternative to Trebuchet MS?
One thing to get out there right away: It looks like you want 1) free, 2) multiple weights, and 3) extended character sets. That's a difficult combination. With a lot of these, a combination of 1 and 3 is possible, that usually means you have to pay to get 2. If 1 and 2 are combined, it's often at the expense of 3.
Back to your question...
Trebuchet is classified as a sans-serif humanist font. Unfortunately for you, Trebuchet is a really unique font. Off of the top of my head I can't think of something that's really similar in the same way that Proxima Nova and Gotham are, or Times and Times New Roman.
FontSquirrel is a great resource for free fonts that can be used commercially as webfonts. The link I provided links you to all of their fonts tagged as "humanist". Out of these, I think Aller gives you the right general proportion and a variety of styles, though the letterforms for plenty of these are going to be different, and there are no Cyrillic characters that I could see. Droid Sans is a humanist font that has a lot of glyphs in it, but won't look and feel exactly like Trebuchet. Same story with Pigiarniq and Source Sans.
Another poster mentioned Google Web Fonts, so I won't repeat the exercise there.
Just for kicks, I ran Trebuchet's glyphs through Identifont to see what it thought was similar. None of these will be free solutions, but they might be better fonts for you to work with. Identifont suggests Prox, FF Plus, Corbel, Elsa, and Corpid as alternatives.
If someone comes up with a great alternative to Trebuchet, then that'll certainly be better than this answer. Otherwise, hope these resources point you in the right direction!