If your board is sturdy but not heavy, wheels under one end and holding onto the other end, or making a grip with some straps, will work for walking.
If your board is not sturdy enough, or what you need to transport is rather heavy, wheels in the middle and a good balance will make it easier.
If you want to cycle with your board you can fix a connector to the board and the other part to your bike, or you can buy a cart with an arm long enough to reach from at least the middle of your board to beyond the end where you connect it to the bike. And like with the walking option, it depends on the board which version will work best.
If the arm of the bike cart comes high enough you can also use it to walk with, giving you two options.
A carry strap to make walking with your board without wheels easier will likely be the cheapest option but it will depend more on your muscles and if you do a good workout on the water, carrying the board home will be a challenge. It might be a fast solution, but I doubt it will be the best in the long run.
Don't worry what other people will think, just see whether you think the road is safe enough to walk and/or cycle.