The best zoo in the London area (and indeed I would say the best zoo in the country bar only Chester of course, which is a little bit far away) has to be Whipsnade Zoo. Getting there by public transport is difficult: train from St Pancras International to Luton, and then a local bus ride to the zoo which if I remember correct from when I tried it two years ago only runs halfway on Sundays. Whipsnade is an enormous rural zoo designed to create large spaces for mammals which weren't achievable at London Zoo, with the historical focus being hoofstock, but in recent years there has been more investment into the carnivore enclosures and the creation of a small but beautiful Aquarium for endangered freshwater fish to broaden its focus a little. If you're prepared to do a bit of walking, you won't find better mammal enclosures anywhere else.
Colchester is another great zoo, quite easy to get to with regular trains from London Stratford to Colchester and then a bus to the zoo which is much more reliable than the Whipsnade one, though again I've only ever done it once. Probably has the most comprehensive collection of big 'ABC' mammals that you could ask for, all with mostly satisfactory enclosures and a lot of theming. The sea lion tunnel, world-class sun bear enclosure and Bearded Sakis are the highlights. Complete opposite of Whipsnade (which is open and sprawling) design-wise, given that it's very compact and small but feels a lot bigger thanks to the clever planting, hidden pathways and extreme hilly landscape meaning that you can never see too much at once and there's a lot to see.
In theory, Howletts should be easy as there are regular trains from London Victoria to Bekesbourne, from where it is just a half-hour walk. That said, as I've never been there by public transport myself, I have no idea how good that walk is or if it's even along pavement. Places like Cotswold Wildlife Park, for example, are theoretically accessible but the walking to get there is supposedly near-suicidal. Howletts is controversial due to the (frankly hypocritical and entitled, I think) views of its owner, Damian Aspinall, but the zoo again has a lot of large mammals in really nice enclosures, and essentially pioneered the currently accepted ideas for how primates, especially gorillas, should be displays (emphasising the importance of foraging and enrichment to counter boredom).
Hertfordshire Zoo was suggested upthread. It has improved a lot in the four years since my visit, which is good news because I personally found it to be incredibly unattractive and a lot of the enclosures really weren't very good. The new Jaguar and Sun Bear enclosures which I haven't seen yet (as well as the tiger enclosure with underwater viewing which I did catch) are star turns indeed. That said, I don't think its do-able by public transport anyway. Liverpool Street to Broxbourne is easy, but from there it's an hour's walk along dangerous country roads; never walked it, but I've cycled it and that was scary enough!
I've never been to Jimmy's Farm, so can't comment there, but I plan to visit soon as it's honestly worth it for the Polar Bears alone (all the other UK holders are very far north). Looking at transport, London Stratford to Ipswich should be easy, but from there it looks like a nightmare to catch local buses half way and walk the rest. If I do go in the coming weeks, I'll probably end up resorting to a taxi sadly.
Shepreth and Wingham are two smaller zoos in the London area that look easy to get to by transport and fairly interesting, but I've never visited either myself. Even smaller is Hemsley Conservation Centre; lovely and easily done by public transport, but absolutely tiny and only really worth it if you want to see Aardwolves.
If aquaria are more your forte, then sadly most of the UK's best are very far from London (Plymouth, Ellesmere Port near Chester and Hull are 'the big three'). Sea Life London, especially considering its location in the heart of the city, is quite mediocre. Sea Life Brighton, however (very easy to get to by train from London Bridge) is considerably better. Founded in 1872, it is the second oldest extant aquarium in the world (after Napoli which opened earlier that same year), and as such much of it consists of tanks embedded into the walls of a beautiful Victorian arcade. As with most Sea Life centres, its exact lineup changes often, but I thought some of the actual displays themselves were also of a higher quality than you'd expect of a sea life centre.
The Belgian Zoos (Antwerpen, Pairi Daiza, Planckendael) are arguably easier than most UK collections by public transport with the Eurostar from London to Brussels, same with (longer journey) Rotterdam and Artis Amsterdam.
As day trips? I've never done it, but I feel you'd be a bit pressed for time. A couple of hours on the EuroStar, and then travelling across Brussels to a different station (at least for Antwerp and Planckendael, not sure about Pairi) and then another hour or so on national train to get to the zoo. All of these are definitely zoos that will take full days, and though I've never visited Pairi I find it hard to imagine needing any less than two days.
