Three seems to be a pretty common number and a reasonably managable group. I got my Gen1 Digimon and my P2 and started them upon getting back into it again after so many years. I started my Jap Gen 3 Digimon (as the others I had were problematic and return shipped) once I received it later. My P2 passed away and I had been running one of my P1's I received since. Currently I only have my two Digis atm, contemplating what I should hatch now.
As far as carrying them around, I am pretty much limited to whatever pockets I have, as I imagine other guys are as well. I usually wear jeans and virtually always have pockets. Back pockets are truly useless for anything whatsoever (including wallets), but I can open up one pocket if my I put my phone together with my wallet/keys in my left pocket. Seriously, who the hell thought a pocket on your rear end which allows you to sit on the stuff you're storing was a good design element?
Anyways, I can put one in my right pocket, and possibly hang one keychain from the small pocket inside the right pocket, if anyone knows what I'm talking about. >_> It works somewhat (prolly because it's very narrow, but you can't really guarantee the keychain will stay hanging there, especially if you're moving around. Might work better if you attach a paperweight or something extra to your keychain to offset the Tama's pull. Like a clip!
Also, my Gen3 Digimon's keychain (it has a clasp chain rather than a keyring) can hook onto a belt loop and hang down into my pocket. So I could have 2-3 in one pocket if I wanted. I've also looked into and invested in some Tama cases, which makes having a couple extra Tama's in your pocket viable.
If you're into hoodies (which I wear pretty frequently), you also have two extra pockets there as well. And when you're into jacket season, that could potentially open up an abundance of pockets as well if you were so inclined.
So altogether, that gives me 2-3 in my jeans pocket (maybe 4-5 if the ones sitting in the pocket are cased, as I don't like the idea of them banging/jumbling around, esp with thrashing keychains), with additional Tamagotchi/pockets if I have a hoodie or jacket. Could also wear one around your neck if you don't mind being a bit more showy.
As most Digimon don't have pause at all and age regardless, I would consider them first to take with me. You can put them to sleep, but it doesn't stop the internal clock aging process (and they will grow and change form if they're chryo-hibernated long enough, which would also restart your clock, should you have it in SET mode for prolonged sleep). Tamagotchi are pausable, and it doesn't effect how much time they spend with you or their growth. Digimon are also easier to care for imo, as there is no discipline to worry about missing, and the Gen 2+ training games are a lot less tedious and quite quick to complete. It's possible to take a bunch with me if I wanted/had to, but on regular excursions, it's prolly better, easier and more favourable to have some on pause while I'm out.
I suppose that most times I wouldn't run quite so many though, maybe 5 max as a 'normal'/'managable' group (5 is probably pushing it, depending what you're running. Some work better than others), so it wouldn't be an issue. Larger amounts would prolly end up in some being paused at home occasionally I imagine, unless I was trying to care for them all simultaneously because someone bet me I couldn't do it.
Or were just up for something different and challenging. Makes it more interesting in terms of what characters you get I imagine. ^^