You can just put ointment on it. Keep an eye on it and if you notice it getting more inflamed and infected, then go to the doctor. Most scratches are okay, it's bites you have to really worry about.
That's almost a daily occurrence at the shelter where I work. We use an antiseptic spray right after being scratched, then antibiotic ointment for a day or two. Unless the area around the scratches becomes hot and inflamed there's no reason to worry.
I get scratched all the time. Those deep dig the claws in scratches that hurt like you know what. I have even had them try to jump in my lap and miss and do the dig the claws in while sliding down your leg scratches that the claws get stuck in your leg and have to pry them loose. Ouch and have been fine.
No need to worry, your scratches look like the I own a cat scratches all of us encounter.