I think they're the best swimmers and potion drinkers in the game, hands down. So play like this. Swim across a lake, jump out of the water, fight something, drink a potion, then jump back in the water and swim away.

I asked a question like this not too long ago, so I'll tell you what I learned from that. I ended up playing almost exclusively Dark Elves after that, so maybe you can make better use of this knowledge than I did.

In PvE, while Argonians sound like they'd be the best healers, it's not quite the case, because Breath of Life crits tend to be overkill in most cases anyway, which can means the healing bonus from the Argonians is wasted, and you'd probably be better off with an Altmer or a Breton for more Magicka. They're still strong healers in PvE, just not optimal. However, with their bonus health, Magicka, and their potion passive and bonus to healing received, they make really good Nightblade Sap Tanks, since they have a good chunk of health (even without investment) and unparalleled resource management with their potion passive.

They really shine a lot more in PvP. Since healing is halved in PvP, their healing passive are a lot more powerful. Health is a lot more useful in PvP as well, and any bonus to resource management is a lot more important, since you won't have anyone running Spear Shards or Elemental Drain for you. Even their poison and disease resistance is very useful in PvP with the Viper/Velidreth meta.

I'd say for PvE, roll with a Templar Healer, a Nightblade Tank, or possibly a Nightblade Healer. For PvP, use a Templar Healer, some kind of Magicka Nightblade, or maybe a Magicka Dragonknight tank.