Adrian,
This falls into the general category of 4-squaring a board, and I'm sure there are excellent videos to be referenced. And probably Marc has a great photo sequence. However, I'll tell you my version, pointing out where things go wrong in my experience......Caution: opinions follow.....
PRELIMINARIES
1. Twist? Check whether the board has twist, as well as bow. If it does, set it aside and plan on cutting it down into smaller pieces. Spend no further time on it.
2. Thick enough? Likewise, if it is your estimate that the board will ju-u-u-st barely meet your thickness requirement by the time you get it dressed, then set it aside, and figure on cutting it in two for smaller parts. Reason: If you think 1/8" is what is needed to straighten it, it'll turn out to be 5/32" sure as little green apples. Nothing worse than over-thin rails or stiles.
3. Bench edge flat? The edge of your bench (where you'll be doing this flattening) has to be as straight as you want this board to be.
WHICH SIDE TO START ON?
4. I like to start on the convex side, because sometimes removing wood at the 'hinge' will cause the board to straighten a bit on its own. But it's easier to start with the concave side up, because it's easier to determine when you have it flat. So to simplify, let's say you start with the concave side up.
SHIM THE BOARD
5. Catch the board _lightly_ between the dogs, and shim the ends so the board will hold still under planing pressure.
CAMBERED BLADE PLANE
6. When there's 1/8" of wood to remove, doing it 1 thou at a pass will have you thinking 'machine jointer' after about 10 passes with no visible result. Start with a jack or fore plane--I like fore planes a lot--with cambered blade so that you're taking off 10-15 thousandths per pass. Go with the grain, so as not to rip off too much (against the grain with a roughing plane is great if you're really trying to do some demolition, but that's not the case here).
7. Work the two high ends separately, being careful to take the same number of passes across the width. Gradually length the cuts as you remove more wood. If there is cup on this side (usually is some), make a series of passes straight across the width all the way down, then 'smooth' the result with lengthwise passes along either side. Then return to flattening, but take no roughing plane passes in the middle of the board--the low place in the middle of the board is the datum that you're going to take the whole face down to, so you don't want to remove material from the low place, other than what is required to correct cup.
SWITCH TO TRY PLANE
8. When you start getting into range, switch to a longer try plane, and make end to end passes to see how close you are to making an end-to-end shaving. Once you can make an end to end shaving, you're done. If the low place in the middle is still, say, 6" wide, then make another set with the cambered blade plane.
CONTROL ACROSS THE BOARD
9. On the first side you do, you can control how much material you take across the width, simply by equalizing your passes across the width. Once you're finished with the first side, you will scribe the thickness around all the board edges, and plane to the scribe line, and that is your true control over thickness in the width.
TURN THE BOARD OVER
10. Having finished one side, and scribing the thickness line around the board, put the convex side up. Take the jack and/or fore plane, and work from the inside in. The board is higher in the middle. Watch your progress against the scribe line around the edge, and lengthen your passes in order to equalize your progress toward the scribe line. When you start getting nervous about the cambered blade removing too much material, switch to the try plane.
The width....This side will probably have 'reverse cup'--it'll be high in the middle across the width. As you begin to get in range on the lengthwise flattening, make a set of passes directly across the width from end to end, removing the reverse cup. Then switch back to lengthwise flattening, to remove the widthwise planing marks and complete the flattening.
As you approach the scribe mark, you decide whether you're going to wait another day, to see if the board wants to move more, or whether you're going to proceed to joinery that day. If it's a rail or stile, I'd wait another day, and leave a bit of the scribe there as a witness. But if its' going into a glued-up panel, I'd plane to the scribe line and proceed right away with the glue-up, and not give the board a second chance to move. Let the glue hold it straight.
Wiley