At this point, most F mount lenses will need to be purchased used, as Nikon has decreased their available new F mount lenses down to just a handful. Some lenses that have been dropped by Nikon are still new in stock at some camera stores, but those stocks are dwindling. The good news is that there are plenty of used lenses for Nikon F on the market.
I can't help you with the portrait lens, but I can make some recommendations for telephoto. I assume that by telescope lens, you mean a long telephoto.
On the cheap there is the Nikon 70-300 F4-5.6. This lens was offered as a kit lens with many Nikon crop frame DSLRs. They can be had for under $100 from places like MPB or KEH depending on specific model and condition. These are not top performers, but offer quite good performance for their price. On a crop frame camera, like your D3400, they will give you an effective 450mm, which is adequate for wildlife photography, although you will likely wish for more reach. The most common models do not have VR, vibration reduction, which is a real minus for this genre of photography.
You mentioned $1000 as your limit. For around that you can get a Nikon 200-500 F5.6. This is an amazing F mount lens for the money. It is very sharp, and has VR. On your D3400 it would give an effective focal length of 750mm, which is excellent. It also has a constant aperture of 5.6. I used this lens for several years, and was very happy with it, although I eventually replaced it with a mirrorless Z mount 180-600 because I currently use Z cameras. You can use this lens with an adapter on a Z camera, if you move on to mirrorless later on. I did that for several years, and it performed well adapted to a mirrorless camera.
You can save a few hundred on 150-600 3rd party lenses from makers like Tamron and Sigma. They are very good lenses, although the Nikon 200-500 is generally considered to be superior, although some may argue about that. They do offer a little longer reach than the Nikon lens.
One thing to keep in mind is that virtually every manufacturer, Nikon included, have switched to mirrorless. No new DSLR models are being developed, and they are quickly withdrawing existing models of cameras and lenses from the new market. If you are going to get serious about photography, you may well decide to move to mirrorless in the future, while F mount lenses can be used on a mirrorless Nikon with an adapter, you will likely want to take advantage of the newer technology available in the Z mount lenses. In other words, if you might switch in the future, I wouldn't spend a lot of money on F mount lenses now, only to end up replacing them down the road. Just my take on it.
