There are many ways to start pepper seeds. I hope this describes the method(s) I have used. These are not the only methods and techniques. There are many ways to do this and depending on your goals for plant size and speed to get it there etc. everything may be different.
I have previously published this post about germinating hot pepper seeds using a paper towel method. I used this most of the time, but realize it is not for everyone.
If you do not use the paper towel method you can sow seeds directly into a seed starting mix. The common part I would recommend no matter what method you use is to soak the seeds first. I still recommend an hour or 2 or at least a good rinse in hydrogen peroxide. After that I now prefer to soak for a day in either distilled water or half a day in salt petre solution and than half a day in distilled water. I normally put the containers in the refrigerator now while soaking. Not necessary probably and not even sure it matters but something I started doing.
Everyone has their favorite starting mix. Some like to use a potting soil. Some like a richer mix with some nutrients. Some make their own custom mix. Many people add ingredients like worm castings. Some like coco coir. There is more choices here than I could mention. I will only talk in detail about what I have had my most consistent success with. You may want to research and find what works best for you. I use this Hoffman Seed Starting Mix. It is a very plain peat based mix. It might even be pure peat for all I can tell. It hasn't let me down, and as best as my memory can serve, I haven't had any seeds dampen off in this mix even though I don't use a fan, and often use a humidity dome too much.
I sift this mix in a colander to get out any larger chunks and twigs. I then add a bit of vermiculite, maybe 1 or 2 parts vermiculite to 8 or 9 parts of seed mix. This I feel lightens it up a bit and keeps it from compacting. I rarely add any nutrients at all with it. I wet it with distilled water or tap water ph adjusted to 6.5 to 7 be for adding to the cells / plug tray. I normally use a 128 cell plug tray but have used other containers also.
I typically don't grow a large number of plants of the same variety. Due to limited space and desire for more selection I often grow 2 to 6 of most varieties I grow. At this point after the cells are filled with mix and packed down I will put the seeds in. If I have germinated using the paper towel method I often will dig small holes to drop the seeds in that have already sprouted. If the sprout has grown like a hook already I will be sure the end of the sprout or tap root end is facing down and the curved part is facing up. Other seeds that have not sprouted yet or if I am sowing directly into soil I will submerge into the soil with curved tweezers about 1/4 of an inch or so. Some people go as deep as a half an inch but 1/4 to maybe 3/8 of an inch has seemed to work best for me. If you are sowing directly into seed starting mix you may just leave some empty space on the top of the cell and drop in the seeds and then cover with 1/4 to 1/2 inch of mix and pack it down. Don't leave it too loose as you will be more likely to get helmet heads where the seed is stuck on the cotyledons.
After this point I will put in a heat mat. If its the first wave of the year an no plants are growing in the tray yet, I will use a humidity dome to help keep the soil moist. You don't want it to dry out too much but at the same time it shouldn't be soaking wet either, just damp. Typically with the seed starting mix I use it drains well and rarely stays too wet.
Now the waiting begins. Some seeds may start in a soon as a few days, others may take a month or more. Ideally the majority of pepper seeds will germinate in 7 to 21 days. Some will be problematic and take longer. Others will be eager to live and be up in 4 or 5 days.
Once the seedlings start to emerge I start with the lights. I keep the lights close within 3 to 6 inches of the starts. As of this point I have only used T8 fluorescent lights for this. I am starting to move to some T5 LED and LED shop lights. I am still a bit behind here. Eventually I probably will be using the LED grow lights but just not quite there yet. I have a mixture of all kinds of bulbs in my T8 shop lights, cold / warm / daylight / 6500k and GrowLux. Since I am just starting seeds and not growing full grown plants at this time I just give a mix of light.
Depending on how many seedlings have emerged in a tray, I will at some point quite using the humidity dome. Too much humidity can sometimes lead to fungus / disease / dampening off. Some people may even start using a fan at some point for air circulation and to keep things from getting too wet. I personally don't use a fan and have not had any issues. I also though have fairly low humidity in my basement and am not growing in a tent or anything.
I try to let the babies grow in this mix in the cells until they get a few nice true leaves going. In the picture below there is a couple plants right in the middle with some nice sized leaves. They are ready to be potted up. Some of the others in this picture would be fine also, but I prefer them to be a little bigger. The roots will be more developed.
Once they get to the needed size I will use a popsicle stick to dig them out of the cells for transplant.
This part often makes people nervous. I often times have 2 to 4 or more plants in one of these little cells. Separating them is relatively simple and safe. I just carefully break up the soil and root ball and pull them apart. Often a few roots will be tangled and broke but I am yet to loose a plant. They bounce back pretty quick as long as you keep most of the roots attached. I have done this with thousands of plants and don't recall every loosing a plant. If I have more plants started than I want I will just discard the smallest or least desirable looking starts.
Here are some more pictures of some 128 cell plug trays full of seedlings.