Just one more thing(actually two, but who's counting!)
1/. lens image quality if usually tied to focal length multiplication. The more focal length range, usually the lower the overall IQ. It's not strictly true, but generally true. That is, a 70-300mm lens usually has less IQ at the long end, than a 150-300mm lens will.
So a 18-200mm lens usually is much much larger to compensate for this loss of IQ at the long end, or it just loses out in IQ terms.
Tamron for many years were the masters of uber long focal length multiplication .. they still have their 18-400mm lens .. recommend to avoid this type of lens for use a the long end!
2/. this is where it gets long and arduous, but it'll be important if you don't stick to recommendations already offered!
AF-S in Nikonspeak has two meanings: Don't worry about the second meaning which related to how you focus with the camera, concentrate on the lens hardware version of AF-S.
in terms of lens hardware, an AF-S lens = AutoFocus Silent wave.
Other lens hardware focus types are:
AF = very hard to find old lenses .. fine to ignore
AF-D = old focus drive type, needs body focus motor
AF-I = very expensive lenses, don't bother researching this focus drive type
AF-S = basically an electronic focus motor drive type
AF-P = Nikons latest focus drive type. Nikon have brought on a whole new meaning of PITA when they introduced this new focus drive type .. be very careful what camera body you choose for this lens type!
AF and AF-D are mechanically driven focus system types. Those two focus types necessitate a camera body that has a AF motor in them.
D3xxx and D5xxx bodies don't have that. These are very old lens types, and not many are sold new. As I currently remember, I think only the 105, 135 mm(DC type lenses), the macro 200mm and 80-200mm are the only brand new Nikon lenses that are body driven(AF-D) that still sell new.
If chasing old Nikon branded lenses, be mindful of the name in the lens and that it doesn't contain AF or AF-D.
What gets more confusing is third party lenses! Basically it's very hard to know which lenses will work on a D3xxx and D5xxx if second hand.
I'm fairly sure almost all brand spanking new thridparty lenses from Tamron/Sigma/Tokina are all silent wave compatible. They have their own nomenclature.
I think Tamron's is USD and Sigma's is HSM.
These are the two telltale signs that the lens will focus on those non focus motor bodies.
So if a Tamron lens is model No. XXmm f/N.N DiII VC USM .. you know that it will work on a D3xxx and D5xxx. If that model was XXmm F/N.N DiII VC only, then it won't autofocus on those two bodies.
This is where the D7xxx bodies(except for the D7500!!) is handy to have.
The D7000 - D7200 all have in body focus motors, so will focus with the non focus motor type lenses!
Some of those old lenses can offer very good IQ/$ value.
So you may pay more in terms of camera body .. you could easily make up for it in cheaper lenses due to the in lens focus type.
if you were to look for a 70-300 lens, I'd recommend a Nikon 70-300 AF-P.
Focusing will be slightly faster due to the focus motor type in the lenses, and according to Thom Hogan both the AF-P models are very sharp at the 300mm end.
I've had a play with the Tammy 70-300 USM lens, it is very large for what it is, and does work well, but focusing was medium slow affair as I remember it. IQ was good, but indoors and dark doesn't allow for proper judgement.
AF-P = a bit of a PITA from Nikon in terms of compatibility.
There is a list of camera models that don't work with this lens focus type, the PITA component is that some camera bodies kind'a work, but not fully, other just don't work at all, and others are fully compatible.
For your purposes, D3000 - D3200 bodies don't work. D3300 and later will.
D5000-51000 bodies don't work, D5200 is semi compatible and D5300 and later work fully.
The breakdown here is, if you are intent on a D3xxx or D5xxx, then stick with D3300 or D5300 and later models.
The AF-P focus drive type is much better than the AF-S on those model cameras for consumer grade lenses.
A nice relatively cheap camera lens combo with up to 300mm would be D3300/D5300 + Nikon 70-300 VR DX AF-P lens, plus a 18-105/18-140 walkabout lens.
But seriously, if your interest is in birds ... 300mm even on Dx is still far too short. Min 400mm(why I recommended the Tammy 100-400 lens) .. 500mm better either Tamron 150-600 or Sigma 150-600 work well at this focal length.
At this stage of your photography experience the DoF equivalence referred too earlier isn't so important.
However! whether a lens is Fx or Dx, makes no difference to DoF! The difference in DoF is related to the framing - focus distance - focal length relationship. Nothing to do with the fact that a lens is Fx or Dx.
Only restriction with Fx/Dx lenses, is that a Dx lens on an Fx body will not produce a full image circle on the Fx sensor. It will produce a very strong mechanical vignette.
Fx lenses on Dx bodies is probably an optimal combo.
Makes no difference if a specific lens type is a Fx or Dx type.
eg. as above with the 70-300mm types.
There must be 3 million Nikon 70-300mm lens models, some Dx some Fx. .. the latest three AF-P lens introduced in the last few years, two are Dx, one with VR one without, and one Fx with VR.
They will all give the exact same FoV(roughly speaking) and DoF for the same focal length set on the lens.
The only difference is that the Fx model is physically a lot larger and heavier. And on a Dx camera, the IQ it produces at the edge of a Dx camera will be of higher quality than the two Dx lenses will give.
note that it's good to see you still 'fishing' .. best way forward is to gather all the info you can before you commit to whatever you decide on.
If it helps too, give us a rough estimate on a budget you want to limit yourself too.
No point in offering $XXXX hardware recommendations, if your budget is $xxx dollars!
OH! and in the world of photography .. the old adage that you get what you pay for is almost 99.999% true!
Pay less, get less. Pay more .. this is where the 99.999% happens .. you usually get more.
So with more of the above now explained, I do highly recommend that camera body should be a minimum D3300, a more better(!!??) .. body would be a D5300 or greater.
If you do want to go down the cheaper lens path at some point in the future tho .. then I'd strongly urge that a D7100 camera body is the way to go.
Some nice fast longish lenses available for that camera body type that would struggle with the consumer spec bodies.
Note that I said 'struggle' and not 'not work'.
Tandeejays specific issue of a non focusing lens on a camera body is hit and miss. My Sigma 150-600S lens focuses fine on my D5500, D70s, D300 and D800E .. just a later lens with better compatibility I 'spose.
Lenses with compatibility issues are usually taken back to service centre formware updated and all is forgotten. I remember this for Sigma lenses was a free service .. almost whilst you wait too.
Some newer lenses, from Tamron and Sigma have this ubute USB dock compatibility that allows you to update firmware in the comfort of your own PC environment.
Also handy to tweak lenses for a bit better compatibility with your requirements.
ps. did I mention the Tamron 100-400 lens!