best tele lens

I am trading my D3200 in for a D5200 and am going to buy a 70-300mm telephoto lens. These lenses all have different letters following the mm rating which makes them differ in cost and I assume quality. First of all, is it OK to invest in a used lens to save in cost? and secondly, what would be the best letters to look for in a telephoto lens and still try and keep it under $200?
Thanks
Rick

Absolutely nothing wrong buying second-hand gear. Just make sure you know what you are buying and the lens ( or anything else ) is in good condition. All my stuff is second-hand and works fine.

As far as letters, I don't know them all. But something like my 55-200mm would be fine which is a SWM ED VR IF lens.
ED = extra low dispersion. Used to correct chromatic aberration.
VR= Vibration Reduction.
IF = internal focusing. Meaning that the lens inside the barrel do the focusing, not the end glass.
SWM = silent wave motor. Just means that the motor is suppose to be quiet when focusing.

Now this is with Nikon lenses. Other manufacturers have their own abbreviations for stuff.

Just remember that the D5200 doesn't have a focusing motor. So you need to buy a lens that has the motor built-in. AF-S or AF-I if you want to auto focus.

There's really no shopping for lenses based on the alphabet soup put on them. They don't make one lens with multiple iterations to allow you to pick & choose which options you want. The letters merely denote certain characteristics about the lens. For Nikon, the letters mean:

AF-I – Internal Autofocus
AF-S - Focusing is driven by a "Silent Wave" motor
A/M mode – Autofocus Priority with Manual override
A-M switch – Automatic – Manual focus switch
AS - Aspherical lens elements
ASP - Aspherical lens elements
CRC - Close-Range Correction system
D - The lens provides Distance information to the camera electronically for more accurate light metering (All G lenses are also D lenses)
DC - Defocus Control
DX - The lens is specifically designed for use on Nikon digital ASP-C DSLR cameras.
ED - Extra-low Dispersion
G – “Gelded”, lens has no aperture ring
IF - Internal Focus
M/A - Manual/Automatic focus mode
ML - Meniscus Protective Lens
N - Nano Crystal Coat
RD – Rounded Diaphragm
RF – Rear Focusing
SIC - Super Integrated Coating
SWM - Silent Wave Motor
VR - Vibration Reduction

The two best 70-300mm on the market are these:
Nikon AF-S VR 70-300 f/4.5-5.6G IF-ED: Amazon.co.uk: Camera & Photo
Tamron SP AF 70-300 F/4-5.6 Di VC USD Lens for Nikon: Amazon.co.uk: Camera & Photo

They both have image stabilisation VR in the case of Nikon and VC for the Tamron. These help a lot shooting at lower shutter speeds and I can say the Tamron's VC is very good as I have one. These both have modern auto focus motors which help with fast quiet focusing.
Another telephoto in that range recommended by some users here is the Nikon 55-300mm. This is a DX lens which won't be a problem if you don't plan on upgrading to full frame. I believe it is lighter than the 70-300 but I haven't used one to know how good they are.

Those are the three I would like at if I were you. This article might also be of interest.Best budget Nikon telephoto zoom | TechRadar

I think Sigma also have an image stabilised 70-300mm, denoted by OS. It might be on that article I posted but I haven't researched that one very much.

Yep, nothing wrong with used at all

Maybe you can get a third party 70-300 for under $200, but I doubt that you can get the Nikon AF-S 70-300 VR for that little. If you can , buy it right away. The Nikon version has been rated as one of the best bargains in the Nikon lineup and can be found refurbed for around $375.

Thanks to all. It is so nice to be able to tap into all this knowledge in the forum! My last nice camera (which I still have but haven't used it in 25 years or so) is the Nikon F. It was so simple yet still took incredible pictures, couple of basic adjustments and away you go! I am sure I will love my new D5200 but not being a techie, I am severely overwhelmed by all the options!!
Rick

welcome!