This is a great way to fish from a SUP. If nothing else it can help you find where the fish are. Floating plastic lures that swim down once under way are ideal. It means you can stop and your lure(s) just floats to the surface rather than dropping into the rocks and weeds and getting snaged. There are many lures to chose from. The main brands such as Rapala with tell you how deep the lure will go so you can chose a lure that will swim at the depth you think the fish will be at.
There is a lot more to it than just towing a lure around. The different lures work at different speeds so you need to find some lures that work at your natural paddling speed. I paddle at about 3-4 mph in calm conditions and I will change lures depending on whether the wind is against me or pushing me along quicker. Also consider that some lures will go deeper if you go faster. So there is a lot to experiment with and that is half the fun. If you want a starting point try the original jointed Rapala lure in 5 or 6cm.
In the summer there can be a lot of weed in most waters, if you pick some up on your hooks you won’t catch any fish. If you use braided line you will see your rod tip vibrate to the swimming rythem of the lure, any change to the rythem indicates you have picked up some weed and you need to check your lure, with normal monofilament line the vibrations from the lure is dampened down greatly. If you are serious about your fishing I would recommend the multi coloured braid. The line changes colour every 10m and makes it easy to judge how much line you have let out.
This would be you paddling to where you think the fish might be and simply casting a lure or jigging some feathers etc. This is a good way to fish for territorial fish such as bass or pike. The advantage is that the SUP board can get you to secluded or remote bays etc where the fishing can be much better.
Fishing at anchor
First of all you don't want miles of rope all over the board so you use a diver’s reel which will hold about 40m of strong chord. Add to that a 1kg anchor and you are in business. Safety note - if you anchor in a strong tide flow be aware that if you fell from the board you could be swept away from your board. You could use a leash just the same as when using a paddle board in the surf.
Freshwater - a paddle board can be use in a similar manor as a float tube but it offers the advantage of being able to see down into the water much better than any other fishing method. It allows to cast your fly into wards the bank, casting under trees etc and presenting your fly in areas where if would be impossible to reach from the bank. A SUP also let's you access the far reaches of large lochs/lakes, getting to areas quickly that would otherwise take many hours to reach on foot.
Saltwater - you can position yourself where the fish are. You can troll a lure until you locate a shoal of mackerel then one located you can switch over to the fly rod for some excellent sport.
Here is a board with everything needed for a full days fishing. Most of the time you don't need all that gear but it can meaan catching a few more fish.