Hi Zenon,
Appologies in advance for the long-windedness and imperfect English.
There are several different ways to do that, with pros and cons of each choice.
What I prefer and do is use friction shifters (my video explaining the
friction shifters pros and cons). They are robust, durable, reliable, and work with any derailleur or number of gears. The downside is they are getting a bit tricky to come by - and the bar-end mounting is not ideal for riding technical (off-road) trails.
A decent middle-ground that I haven't tested personally are the
Gevenalle levers which provide a "normal" mount, but can take friction shifters.
The more "standard" approach would, of course, be to get modern brifters (the ones that aren't as durable and aren't built to be repaired). I love the ergonomy of Shimano Claris brifters. Yes, the 8-speed stuff. They shift well and precisely. The 8-speed cassettes and chains are cheap, and Claris comes with 3x8 shifters (you can use the triple front shifter with a double FD and cranks, but you can't use a double front shifter for triple cranks, so I see that as having an extra option if needed). 3x8 provides a huge gearing range on the cheap and I like that.
For an even more "standard" approach, I would look for Sora brifters (2x9, or 3x9). Sora stuff is good quality.
If you wish to further "update" the drivetrain, there aren't many options I would call reasonable (unless you really want just 1-tooth difference across at least the "faster/smaller" half of your cassette sprockets). The new Tiagra 4703 would require new derailleurs (both the front and the rear), and a new 10-speed cassette and chain. The current 105 and Ultegra are compatible in terms of derailleurs, but they work with 11-speed cassettes and chains, and don't offer any triple shifters.
Shimano GRX comes with some interesting choices. A 46-30 chainring crank, that should work fine for most climbs and flats, while being "only" double (IMO that is better than having a 50-34 crank). However, it is designed for
wider ("boost") hubs. So, you'd be running with a less than ideal chainline on your Bianchi (though, when using 46 instead of 50 T at the front, you would be using smaller, more outwards cassette sprockets and that would compensate for the chainline to a great degree - in practice).
GRX comes in 10 and 11 speeds.
All the above-listed solutinos are roughly double the price from one to the next (it adds up, and the "higher end" components come with a price premium, and a slightly lower weight - other differences are debatable).
Microshift stuff is OK. I would say it's not up to the Shimano mid and high end quality, but it's far from what I'd call bad (though Shimano is no longer what it used to be in terms of durability I'd say).
I hope this gives you enough info to make an informed decision.
Relja