The first step I would take if I were you is a review of available calculators that perform the functions you want your calculator to perform. Then I would compare their selling price against the cost of building your own calculator. You will probably be able to cannibalize an inexpensive calculator to obtain parts, such as you illustrate in your very nice photo, so the really expensive part will be the design and construction of a circuit board that ties it all together: microprocessor, keyboard scanner, display driver, battery monitor, maybe a battery charger if your want to operate it like a cell phone with rechargeable lithium-ion cell(s).
To get you started, I found
this spiffy little programmers calculator made by the Japanese firm Casio Computers Ltd. with a suggested price of US$14.99. The clever engineers at Casio threw in a few extra features that you can examine by downloading the Manual and Additional Functions PDF files from the links on the web page.
Their calculator doesn't need recharging from a wall outlet; it uses a solar cell array and an internal battery. I've used Casio calculators for about thirty years now and always found them to deliver good value. Of course, i still have no idea how to program a microprocessor to perform the things a hand-held calculator does, but I do know how to program a microprocessor to do all sorts of things that a hand-held calculator simply cannot do. I suggest, if you want to learn how to program microprocessors, that you start with any of the many microprocessor hardware and software development kits. These are incredibly affordable for the DIY electronics experimenter.
On an historical note: I am fairly old. I started programming microprocessors for embedded system designs in the 1970s using the Intel 8080 8-bit microprocessor. One of the first things I programmed (just for fun, it had nothing to do with the project I was working on) was a decimal calculator that would add, subtract, multiply, and divide absurdly large decimal numbers. The 8080 happens to have a BCD arithmetic function that allows two unsigned BCD numbers, packed into a single byte, to be added or subtracted using
two carry bits and some programming logic. Once the programming for that is nailed down it is almost trivial to dynamically extend the number of digits to any arbitrary length. Multiplication and division are more difficult since these functions have to be programmatically implemented. Early microprocessors had no multiply or divide instructions, meaning they are slow to multiply and divide, unless you forked over money for an arithmetic co-processor (which IIRC wasn't available for the 8080).
Still, it was a lot of fun and a good way to learn assembly language. Today we have PICs, FPGAs, and 64-bit microprocessors to play with, along with gobs of free (or almost free) software in whatever language you feel is appropriate to the task at hand.
Please keep us informed on what you decide to do with your project.