In comparison to today’s telescopes, how powerful was the one(s) Galileo used? Or was it more on terms of today’s binoculars?
I’m not too sure, but this site claims to be concerning a reconstruction of Galileo’s “14powers” telescope.
Could be comparable to powerful binoculars today.
Here’s a museum piece said to be one of Galileo’s scopes. It’s 14 power, f/51, with a 26mm aperture. Typical binoculars are 7 power with a 50mm aperture, which would have half the magnification but four times the light gathering ability. There are binoculars that could outperform his telescopes. 
In the third paragraph of the Sidereus Nuncius, he claims that the telescope he’s been using has a magnification of 30X, but this is believed to be an exaggeration. That’s what he’d expected the magnification to be and it was difficult for him to actually measure it accurately. In his paper “The Telescope in the Seventeenth Century” (Isis, 1974, 65, 38-58), Albert van Helden noted that:
(But I note that, in his footnote, the reference by Ronchi and Abetti are dated 1923. Hmm. Though he cites a later book by Abetti as well.) The biggest difference with the average modern pair of binoculars is however that all his telescopes had a terribly narrow field of view. For the objective above, that was just 15 minutes of arc, about a quarter of the Moon. That’s very impractical for terrestial use, a problem that wasn’t solved until about 1645.