Using plain screws into just about any material other than wood or metal will not provide a strong support, especially if the weight that the screw is supporting is trying to pull the screw shank straight out (rather than pulling across the shank of the screw).
From your pictures, the surface appears to be plasterboard (or less likely, plaster), which has to be attached to a support structure, either wood, steel or concrete.
Wood
To test if it is wood under the plaster/plasterboard, remove a screw and insert an awl until it just stops.
![awl](https://i.stack.imgur.com/ymL0x.jpg)
With some force, try to push the awl in further. If it is wood, the awl should sink in a bit, but not too easily. There should be enough wood to allow you to drill into it and seat the screws fully. You probably need at least 1/2 inch of screw thread in wood for a safe hold on blinds. More is better. Remember that the plasterboard/plaster doesn't count. Try to guage how thick it is (probably either 3/8 or 1/2 inch) and add that to the depth into the wood.
Steel
If the awl doesn't sink in, it may be a metal stud or a metal beam. If you hit something hard and the point can slide easily side to side, it is probably metal. Try to drill a small hole through it with a sharp drill bit. If it is a steel stud, you will be able to with a bit of pressure. You can then use a sheet metal screw to hold the bracket up. If it is a steel beam, you will not really be able to (it is too thick and hard). You will need a whole different mounting system.
Concrete
If the awl stops but can't be slid along a hard metal edge, it may be a concrete support structure. The awl should feel as if it is digging into something a bit gritty. You can mount into concrete but need to use a masonry anchor.
![lead shields](https://i.stack.imgur.com/tmoOf.jpg)
To insert the anchor, you first need to drill a properly sized hole with a special masonry drill bit.
![masonry bit](https://i.stack.imgur.com/ARfPu.jpg)
It has a carbide insert in the tip. It is slow going and should be much harder than the plasterboard/plaster. The dust probably will be greyish or tan in color. Be sure not to drill a hole too close to another hole, or the concrete between them may crumble. You then need to insert the anchor and screw the bracket into it.
While it is theoretically possible to get a good enough mount in concrete like this, it is hard. This is especially true when the load is dynamic (moving) the way blinds are.
Plaster/Plasterboard
While it is possible to hang light to moderate loads on plasterboard or plaster, it is really only safe when it is a vertical surface, not an overhead one. You can use various anchors, but not when the force is pulling directly out from the surface rather than down along the surface.
There is much info on how to mount blinds on the internet.
This type of blind USUALLY mount inside the window casing ,to the top of the window casing which is typically wood. If so then 1" wood screws would be sufficient to deal with the weight of of the blinds and the mechanical forces of pulling them down.
BUT the photo on the IKEA site seams to indicate that the bracket is mounted to the wall, If this is how you will be mounting them ( and your wall is made of drywall ) then you you need to know if there is structure in the wall at the mounting location. If there is structure, stud or header etc, then any good wood screw ( at least 1 1/4" ) will work. If not you will need a toggle anchor to deal with the forces involved.
I prefer Toggler anchors.
As manassehkatz says, you need to know what your wall is made of in order to know how to proceed.
Best Answer
While this question may be ambiguous or style based I will try to give the best answer I can and let more qualified hands cast close votes.
I would recommend sheers. They allow plenty of light in, but obscure the inside of your apartment enough to allow some privacy. In terms or hardware, they are extremely easy to set up, if you have a recessed alcove window, just an expanding dowel rod is needed. However if not, you will need brackets, this is also true for all blinds and curtains. I find curtains and blinds either bulky or they close out too much light when closed and offer no privacy when open.Sheers are the happy middle. If you want even more privacy then the link shows, a easy to add more is to double the sheer up. The obscuring feature of a double sheer is compounded and yet still allows ambient light in while providing absolute privacy. However in the bathroom and bedroom, its possible you will still feel more comfortable with a thicker blind in which case a underlying sheer with a solid curtain over it is best. To maximize the sheer, tie back the solid curtain during the day.
In terms of absolute style, which this site isn't about, the cream color of the magnolia and the brilliant white of the sheers is seen in nearly every home in architectural digest or living magazine. Also white allows you to use varying colors in the room as white nearly goes with everything. All attempts at trying to use heavily colored sheers or drapes have come out looking Victorian and heavy. Plus choosing too bold a color as a big pieve of wall space sets you up for disaster. Depending on the wood color you could very well get away with a cream color sheer.