Birds mainly, though not limited. General landscape photography now and then. I live right by RSPB HQ and several reserves and lakes on my doorstep. For instance, I took my 1 year old out for a walk on Sunday afternoon around 4pm and we watched barn owl getting it's hunting on early. We get a lot of wildlife by our house, mostly nocturnal though, but that's another matter entirely.
I was thinking second hand to start for sure, just something with at least some reach (doesn't have to be a massive lens) to begin with to get me going.
Only downside is that you are going for the two extremes of photography, wildlife and landscapes...As said I like Olympus cameras so I'd be inclined to see if you can stretch to something like a second hand EM1 mk2 and either a Panasonic or Olympus 100-400 lens secondhand to cover the bird side of thngs, you could go for a bridge camera but generally they have smaller sensors so will struggle more than the M43 sensor cameras (Olympus/Panasonic) in changable light. To give you some ideas of capabilities of the em1 mk2 I think there are a few of my shots on the thread Wildlife, Animals, Birds, Zoo (barn owl page 75, Osprey as well) - though later shots are with the newer OM1, links through to my flickr account as well.
To give an example for an owl in flight I'd be trying to get the camera to achieve around 1/800 or 1/1000 second minimum shutter speed, easy in summer and good light not so easy at current time of year at 4pm onwards (well not where I live anyway). To get suitable detail of the owl you are needing to get reasonably close to the subject - it's not often you can do this at an RSPB sight as they tend to want to keep you at arms legth also not sure a bridge camera would be capable to do this either, well not easily.
Landscape photography, I don't do much of this other than occasional snap on holiday and I use a Olympus 7-14 lens or 12-40 lens.
It's hard to give a complete answer as for me what I want from an image or deem acceptable from an image might be not good enough for you or way overkill for you.
Only thing I will add my 11 year old daughter got a camera last year, a compact she got some ok pictures with it but was frustrated that when the light was less than ideal she struggled to get any decent shots, she now has an Olympus EM1 and uses my old panasonic 100-400 lens and is more than happy with the vast majority of her shots - I think that camera was around £200 secondhand from Wex.
Hope that helps a bit and doesn't confuse too much, anymore questions just ask.