Red Flowers can be found in birch-like Ashenwood-trees growing on their levaes. These flowers are blocks that can be placed and even used as building-material. They glow very faintly, so they cannot be used as a light-source.
Red Flowers can also be occasionally discovered in Wood Treasure Chest.
Ashenwood trees can be grown from Ashenwood-Saplings (an additional harvest when gathering leaves), but only in fitting environments. Grassland is not always a good place to grow them. You will be know if the area is fitting as soon as you have placed the Sapling. If it stays "fallow" (visible written in red color when you point your crosshair cursor at the Sapling), then you can just take it up again and try planting it elsewhere.
Tree Saplings can be grown on Grass-blocks on surface Biomes, and unlike the hints that you can find ingame, applying moisture is not even necessary. Saplings can be fertilized by Pigsy Droppings, but only if the Saplings aren't fallow, because in this case no tree will grow from them at the chosen spot anyway.
More Red Flowers will - slowly - grow on Ashenwood Leaves with time all by themselves too. So you can lay out flat artificial biomes made from Ashenwood Leaves and wait for a few (real time) days until you can collect one or two Red Flowers there every couple of days or so. The same goes for Blue Flowers, Yellow Flower and Wildwood Flowers; not for Autumnwood Flowers nor Weepwood Flowers though.
An even faster method to get more Red Flowers is to place blocks of Red Flowers on the ground and pick them up again - once in a while you will obtain 2 flowers at once when doing that. This is also possible with Blue Flowers, Yellow Flower and Wildwood Flowers.
Red Pigment is used for crafting Red Beacons, Red Lamps (Wood Burning Lamps emitting red light), Red Fireworks, Red Stained Glass, Red Beds, Orange Block of Goo, Red Concrete, Red Mosaic Tile Wall, Red Adobe Floor, Red Adobe Roof, Red Adobe Wall, Red Carpet and Red Wood Wall.
Red Flowers can also be used as an ingredient in Basic Health Potions, Stone Mining Cells, Taming Collars, Golden Potions and other crafting recipes where you can choose from different possible flowers.