Personally I tend to use my ears, but I have a lot of birds around me and existing audio recorders. I'm always concious that by limiting my hearing to listen to a specific bird I might be missing something else.
Now before we get into any of this - you need to decide on a microphone, this is your most important purchase and applies to and can be used on just about anything that has been discussed. Skim through the various threads to get a feel on what might appeal, my guess is that something fairly manageable to hold would be adequate as it sounds like most birds are nearby. In addition its connection requirements, be it phantom powered xlr or 3.5 mm jack will help dictate what is easiest to use as a headphone interface.
As you are using headphones for long periods check hi-fi reviews to see if they are comfortable for extended listening.
One you have decided on the microphone we can come up with a way of connecting everthing together that suits your needs, there are many options I have personally used, and lots more that I have not, so believe me that should be a doddle.
N.B I don't use bluetooth so will never comment on it.
There are a number of computer usb interfaces that will take a standard microphone input, (xlr , 6.25mm jack ), and use its inbuilt pre-amp to raise the signal to a sensible level and let you monitor the input with headphones. The interface gets its power from a laptop usb port, so you can hit record if there is something exceptional you want to record. I use a Focusrite Scarlett device as it has optional phantom power and direct input monitoring(needed to listen directly to your microphone without it having to pass it into the computer. Ineviably these are intended for music production, and the instruction book is overcomplicated for bird use. Instead of the complex DAW it will work quite happily with freebe software like Audacity. The added bonus of this type of approach is that it by-passes your computer audio card giving you better reproduction.
This is one of many approaches and I would rather get the microphone nailed first.
As with most things audio, once you have decided on what you are about to do, put up the details in this section so that someone can make sure there are no hidden flaws and give a bit more precise guidance on getting it up and running quickly.
As you can see Andrew and I are saying much the same things which hopefully is reassuring.
J