In iTunes, go to your playlist. On the playlist, click on its column header row at the far left end, above the column with sequential numbers below. That sorts the playlist by playlist order, and you can change the song order by dragging songs around on the list. You may have done this already.
If you sync this playlist to the iPod, it should appear in that order on the iPod.
If it does not, you may not be looking at the playlist on the iPod. If you select the list of All Songs on the iPod, it will be in alphabetical order. Even if you only have one playlist synced to the iPod, the All Songs list is not the playlist. If you select the playlist on the iPod, it should be in playlist order.