What type(s) of wood would be appropriate for this use
The usual answer to this commercially from back in our parents' day to recently has been one of a number of tropical hardwoods, including teak in the past, and more recently iroko and ipe. For many teak has now become unobtainable, or prohibitively expensive, and iroko is not much cheaper. And the price of ipe is certainly not dropping any time soon.
Anyone living in the west should have access to native woods that are naturally rot-resistant as there are a few species with high exterior durability growing in all parts of the world.
For North America you're in a lucky position as there are a good number of options, see this table from the Forest Products Laboratory showing the relative decay resistance of commonly available woods:
Any wood from the Resistant column will give the kind of durability wanted, and even a few from the Moderately Resistant column can prove to be acceptable.
Not just natural wood
In addition to natural woods there are now treated woods such as Accoya which offer the same sort of durability. Cost and availability can be a problem, but they do come with a guarantee which other woods do not!
how should I treat it, or seal it against the elements after I replace the slats?
With species that have high natural resistance to rot you don't have to do anything to them for them to endure outdoors, this is one of the reasons for using them in the first place.
On boats and for garden furniture teak was traditionally often left bare, to weather to a silvery grey. But not everyone likes this look and would prefer their wood to keep its original appearance, which led to the development of "teak oil".
Today, while "teak oil" is still widely sold1 there are various other treatments and coatings available which provide varying levels of protection and a range of looks, including penetrating clear wood sealers at one end and CPES followed by marine varnish at the other extreme. And of course there is paint, which has always been available to provide excellent protection to wood.
If you are OK with finishing....
It's important to realise the implication of finishing products providing good protection from the elements even to common softwoods. If you are comfortable using a given protective finish (not just at time of build, remember there will be an upkeep schedule2) it opens up your choice of wood to nearly anything that meets your other requirements — cost, stiffness, looks.
1 More than a dozen brands are available in the US although it's important to note that they are by no means all the same finish.
2 From yearly at the low end to a maximum of about 5-7 years for the majority of consumer-level finishes.