The Lambi area of kos is delightful.there is a nice beach with a relaxed atmosphere, plenty of good restaurants and kos town on your doorstep if you want anything livelier.

In May most of the islands are going to be quiet, with the possible exception of Santorini. We have visited the 3 islands you mention and all have resorts to suit although the beach in Molyvos is not very good. Visiting other islands is easy from Kefalonia & Kos on tourist trips but from Lesvos its a ferry from Mytilene to Chios, not so convenient. Drunken Brits tend to stick to one or two specific resorts on many islands and this tends to give the whole island a bad reputation but in reality these islands are fine on the whole, for instance Kavos does not ruin Corfu. In May you are likely to be encouraged into tavernas but although it can be annoying if you just want to walk around and choose for yourself its good natured and 'tomorrow, maybe' does the trick. Lesvos is our personal favourite but to make the most of it you'll need a car, its a big island and it took us 6 weeks to get to see most of it without rushing.

Kefalonia is a wonderful island and it as rdv says, the neighbouring islands (Ithaca, Lefkas, Meganissi and Zante) are relatively easy to get to.................BUT, you will need to hire a car to do this unless you take an excursion. Public transport is virtually non existant on Kefalonia. There is only one bus running from Skala in the South of the island along the west coast to the capital, Argostoli. This leaves Skala at around 0845 in the morning and the bus leaves Argostoli at around 1415 heading back to Skala. This means it is impossible to see anything of the island without a hire car.
Kefalonia is a beautiful peaceful island and you would never find a rowdy bar, everything is laid back.
However, have you also thought of Skiathos as a destination? All of the beaches are sandy but i would recommend Koukanaries. It is at the furthest point from Skiathos town and is very laid back with a pine fringed beach which is very safe for swimming. Buses run into town every fifteen minutes and there are also water taxis to take you around to the less accessible beaches from Koukanaries.
There are also lots of boat trips to the neighbouring islands of Skopelos and Alonnissos (and smaller ones too) from the old port in Skiathos Town.
I really think Skiathos and Koukanaries would fill all the criteria for you.

Good luck!
John

Thanks everyone for your advice and suggestions.
We did look at Skiathos but unfortunately we have no flexibility about what date we travel due to work, and there are no flights to Skiathos on the date we can go. I think it's definitely going to be either Lesvos or Kefalonia, and although we haven't decided for certain yet, we're leaning more towards Lesvos as neither of us have been to the Aegean islands before, whereas we've both been to Lefkas and my friend has been to Zante so we're not sure if Kefalonia would be fairly similar. It also looks like it'd be easier to get around by public transport in Lesvos.
Can anyone comment on things they'd recommend doing on Lesvos and Kefalonia?

Most of Lesvos is not accessable by public transport. The one exception is the excellent seasonal bus service that runs frequently between Anaxos, Petra, Molyvos & Eftalou. Driving is very easy apart from Mytilene which is chaotic in the morning. The roads are very good and quiet out of the towns

For the reasons rdv gave above I'd look at Anaxos/Petra or Molyvos/Eftalou (which has hot springs to visit). Molyvos itself is very picturesque indeed and not too resorty, but has not a great beach in itself. Anaxos/Petra are near each other with nice beaches, and decent bus link into Molyvos (later in the evenings it's a taxi job though for several euros I guess Molyvos-Anaxos)) . Otherwise, getting around the island without a car is really a difficulty. Google picture-search the places mentioned to see what you like the look of. Molyvos is lovely by night and has cobbled,winding streets up to the castle with places to shop/eat/have a drink (it doesn't have a wild nightlife). It has character by the bucketload - but you need to like a bit of uphill/downhill walking! A bit more on Molyvos
http://www.lesvos.com/molyvos.html
and the hot springs
http://www.lesvos.co.uk/thermal%20springs/eftalou.htm

Cephalonia isn't easy to do without a car either in the main. Lourdas has an immensely steep walk to the beach if your accommodation is near the top, but a nice place! Lassi is more resorty/caters to the British, I guess (but certainly not full of drunk Brits), but if you want to get into Argostoli by day/evening it's possible to walk, and not a far taxi ride. It has sandy, but not huge beaches.

No hassle really in either of the above to go and eat in places.

I can't comment on the places you've mentioned in Kos. But of the other two islands/places you've mentioned (and given you want to get out-and-about somewhat), I'd investigate the Lesvos options more closely. Sorry for the rambling!
A bit more on this subject here
http://www.tripadvisor.com/ShowTopic-g189479-i1459-k3468320-Molyvos_or_Petra-Lesbos_Northeast_Aegean_Islands.html

I notice you don't mention Lassi in Kefalonia, a beautiful resort, and oh, so convenient for travelling.

We loved the fact that the airport is just a few miles away from Lassi, so no costly transfers, just a small taxi fare, and you can be in your apartment or hotel within half an hour of landing.
Also, Lassi is just 2 miles from Argostoli, the capital of Kefalonia, a short taxi ride, or a very pleasant walk.
Beaches are not brilliant in Lassi, but we're not 'beachy' people, so it didn't matter too much to us.

We had a wonderful, peaceful, relaxing holiday in Kefalonia, just wish we could say the same about Turkey this year. Serves us right!! Greed took hold, and we thought we'd get more for our money in Turkey........... Wrong, we spent a fortune, and weren't impressed with Turkey.
Lassi next year........... Definitely!!

I've been to Tingaki in Kos although it was a good few years ago. If I can remember correctly you could get a bus to nearby Kos town. It has ample bars and restaurants without being over the top and in your face. It also has a long stretch of beach. We hired bicycles as it was nice and flat.

I've been to Katelios,if you're a grecophile then go,it's lovely and the view from the eachside tavernas accross to Zante at sunset are worth paying for,there's not a lot going on in the village in the daytime but the evenings were lively enough if you want to just sit and watch the world go by,have a meal etc but i would imagine may would be fairly dead and as said,public transport on some of the less developed islands can be a real hurdle,kefalonia falls well into this category but hiring a car is cheap enough. i don't know if it's available anymore as a package but i stayed in the Hara apartments,they're out of the way with a lovely pool in a rural setting,the owners are the typical welcoming locals that make your holiday memorable,but it didn't have any facilities such as a bar or anything,lovely rooms though with aircon and what not.

Kefalonia is breathtaking. I would second the previous post re Katelios- we didnt stay there but went there for many a lovely evening. You will need a hire car though. They were building a lovely hotel there with a nice pool when we were there this summer.