Heya,
So a budget of under $300? What's the highest you can go? I would suggest saving and getting something very functional and able to expand in the future as you develop more skill, experience and your own style. Used will get you further with a budget. I would shoot for a dSLR if you want to have a big range of options. It will cost a bit more than point & shoots. You also have the learning curve of using a dSLR and properly setting up shots (not just using auto). Banking $300ish for a camera and another $200ish for a lens would be a safe bet.
Sounds like you have three different things you're shooting. Lanscape. Wildlife. People. Those require three different focal lengths. Landscape, wide angle (on a crop sensor, that would be 10mm to 18mm). Wildlife, telephoto (200mm to 400mm as a starting point, if you want to fill the frame, if you're ok with 20~25% frame fill for a heavy 60% cropping, then 200mm can do that, this will be your biggest challenge on a budget). People, anything really, but to be safe, 35mm to 135mm is a very wide range depending on how you are shooting people, but something in there is likely to be your zone.
You could get any basic dSLR camera for this. A simple used crop sensor will be the most inexpensive way to go. Or you could go new on something like a SL1 for example, as it's not that expensive. Then add lenses over time. I would also just look at the Canon T3
, which is $279 shipped right now (body only).
For lenses, you could use basic prime lenses. But an entry zoom with a broad range will likely give you your common focal lengths for less money so you can get to shooting right away with less lens swapping.
Tamron 18-200mm is a great little zoom and very inexpensive. This covers the wider angle and the portrait zone and the telephoto zone. Would be a great entry point. Sigma has an alternative to this lens too. Canon does as well but it's quite a bit more expensive. Here's the Tamron for $170 shipped.
Or maybe a Sigma 18-250mm with OS (stabalization) for $300 shipped.
That'll get you started pretty quickly. Then learn photography well with that. And from there with some experience, save up some budget to specialize your setup for what you know you need after you learn to use what you have.
Other accessories to look into:
Extra batteries.
A manual flash + batteries.
Tripod/monopod.
Remote shutter release.
Used equipment that is in good shape will get you further per dollar.
Very best,