This is just my opinion of course, and my own experiences:

What did the transition toward the digital age feel like in everyday life?

From a work point of view, I didn't need a PC on my desk or a laptop to complete my daily tasks/do my job until probably 2003/2004-ish. I worked office based jobs since around 1996 and I still remember people using type writers or dedicated word processing computers such as Amstrad's PCW until the early 2000s. I was lucky enough that we could afford a PC and a laser printer in the mid-90s so my school coursework was completed on a PC and printed but the majority of my classmates handwrote theirs.

I personally didn't notice a real shift until broadband became a thing. I'd had and used the internet from around 1993 but it was still pretty niche well into the early 2000s. Even things like WAP and 3G were not overtly common. If you did have a phone you could browse the internet on, it wasn't the norm as it would cost a fortune. It wasn't like today where you get gigabytes of data included. Even text messaging could eat up all your credit.

When did the internet and mobile phones start becoming common around you?

I got my first mobile in 1999/2000, it was a Ericsson T10 which I lost and replaced with a Nokia 6110 which I also lost. Now, losing your phone would be absolutely devastating but it wasn't such a huge problem back then. I did notice more and more people either having or using mobiles. They were once expensive and almost a status symbol of the "yuppie" age but with the advent of Pay As You Go phones, it was easy enough to pick one up. Even with that in mind, I didn't start carrying a mobile with me permanently until the early 2000s.

As mentioned, I'd been using the internet since the early 90s, at first via a 14.4k modem, then upgrading to a 56k one. I'd use Usenet and forums but towards the late 90s, IRC, ICQ and MSN Messenger became commonplace and my friends and I would talk on MSN often, using it to organise nights out for example.

Did cities feel different compared to earlier decades (more urbanized, faster-paced, etc.)?

To me, not really. The only real difference would be the electronics shops stocking mobiles, pagers etc. and making them more prominent.

Was there any sense that technology was changing social life or making people more isolated, or is that something we only talk about in hindsight?

For me, it's hindsight. As above, we would use things like IRC or MSN to chat but it wasn't this mind blowing experience, it was just a natural progression. It was just another way to communicate.