I want to upgrade my 2020 Allez from Claris to Shimano 105 I have a 2020 Specalized Allez that came with Shimano Claris, but since I live in a very hilly area, I feel as if I need more gears. I have been biking for a while now, but I don't know if I need to train more or need more gears. The inclines that I find myself needing more gears on are over 10%. I am wondering if I can just put in a 105 rear and front derailleur, and keep my original brakes. I am also wondering if I have to get new rims to compensate for the extra 3 gears I am putting in, or if I can just put a more narrow spacer in the back.
Thanks for reading!
Edit: For those of you wondering, I have the base claris gearing system, so an 11-32 cassete in the back and 34-56 in the front. Thank you all for your help. When going up 5-12% inclines, I don't need to shift all the way down, its usually in-between 13-25% inclines that I find myself using my smallest gear. I will probably train more for the next month or two and then see if I still need to upgrade! Thanks everyone.
To go to 105 you would have to do the following:

Front + Rear Derailleur
Cassette, Chain Rings
Chain
STI Lever (at least the right one) - This alone is like $150 per lever if I recall correctly.
Possibly new rear wheel

Overall most people will not make a jump from Claris to 105 because the costs usually end up being almost the same price as 50% or more of the value of the bike.
Now 105 to Ultegra/Dura-Ace is more common and cost effective (e.g. As I wear out components in the drivetrain I am replacing them to Ultegra).
What you can do or at least what I would do is this:
Check your current number of teeth on your largest gear on your cassette and count the teeth on your chain ring. If you are struggling on climbs I would try the follow in this order:

If your cassette is not 11-34 Claris I would spend the $20 or so and try that: Link
If you are still struggling on climbs replace your chain ringings to 46-34T. This will not necessarily make the climb easier because you still have a 34 T chain ring and max 34 T cog in the back, but it will make flatter portions feel easier, which can make the climb feel more doable: Claris Chain Ring
Save the money you would have otherwise spent on a 105 upgrade and use it towards a full carbon bike with 105 or Ultegra (e.g. Trek Domane SL5 and I think Specialized Roubaix not sure of which trim though).

Here is the important thing to realize though it will not get easier even with 105 or another bike (assuming both bikes are of equal weight/aero/etc.) if your cassette already is 11-34. The reason being is that 34T is the max size rear gear you can get whether it is 105 or claris and the smallest inner chain ring on both sets is 34T.
Also, are you sure the gradient is 20%. That is pretty steep for a bike - Link. Many cyclist will have to walk that:

                
            
Here is what 7% or so looks like as a reference near the end:

                
            
In that clip, I am in a 34 T chain ring and the 4th cog of a 105 11-34 cassette. So to your other point with training climbing 7-10% is possible even for an amateur.
Hope that helps.