You don't need any special tackle to be able to fish from your SUP. The most basic tackle would be a simple hand line and a few mackerel feathers, read on for guidance on the options.
Rod and reel
As has been mentioned you don't even need a rod and reel a simple hand line can suffice but using a rod and reel makes things more fun.
A cheap spinning rod and reel serves well for 75% of all SUP fishing. You generally don't need anything heavy so a rod rated to cast up to 28g will suffice . One thing to consider is the length of the rod. For general use I find a rod of 6 to 6.5 ft works well. Longer rods are problematic when weed etc tangles in the tip ring as it is hard to reach the tip to sort such things out. You can go for the very short rods designed for kayak fishing but there isn't really any need to go much shorter than 6ft.
The rod can be matched with a small fixed spool (spinning) reel and line of about 10lb.
On a general note - bare in mind that there is a chance you might drop your rod and reel over the side so don't go taking an expensive rod and reel out until you have assessed the likelihood of loosening it over the side. Start with a cheap combo and go from there. Another consideration is that your equipment is quite exposed on the SUP and gets wet. Fishing reels can quickly corrode and start giving problems. Personally I like using small baitcasting reels and some of them can suffer from corrosion even though they have stainless steel bearings so if you can, go for a reel that is rated for seawater use from someone. I use the Shimano Calcutta reels but they are expensive, the Cardiff range have the same saltwater proof bearings and cost a lot less.
Fly rods - slightly problematic if you get a tip ring tangle but fly fishing is certainly possible and the board offers excellent possibilities for 'sight fishing'.
For anchoring techniques a light boat road may be more suitable.