Steve, I do angling photography for a living so may be able to help
To start with, see how you get on with your existing lens, the one that came with the D3100. That focal range is fine for catch shots, be it holding shots of stuff like skimmers and tench, or the knelt-down poses you need when holding specimen fish. It's a new camera so you can play around with higher ISOs to make up for the fact the lens doesn't have a really fast maximum aperture. See how you get on....
If you do decide to move up to something faster (aperture-wise) and with better optics, the suggested Tamron is a good choice; I used one for a few years before I moved over to the Nikon 17-55mm and TBH, it was brilliant. To kep the price down, go for the DiII version. It's the one without VC and is older and cheaper - plus there are plenty going used generally. It's arguably better in the optics dept. than the newer VC version...
Primes are great - lots of specimen/carp boys use 35mm and 50mm primes - but if that's going to be your only lens, it's a big choice to make because you need the room to shoot, not always possible at many fisheries, plus if you think you'll be shooting at f/1.8 all the time while holding a big fish, think again - DoF just won't be enough to give you a safety net if you're not set up 100 per cent correct. Probably the only times I'm shooting at f/1.8 is to get atmospheric shots with lots of background blur. Even f/2.8 is an aperture I generally only use in low light circumstances or where I want to overlay text.
There are loads of other zooms too that people go for, but I think a 17-50/55 (or what you already have) is the best option for versatility
Off-camera flash is a good idea, as suggested, but then you're going into the realms of doing a photoshoot and if you're just out for a day's fishing, it's more gear to carry (and I know we already carry the kitchen sink!!!!)....
Hope this helps - anything else feel free to ask