I believe this can be a good first learning project, but it is non-trivial, and you will have to learn quite a bit along the way, as well be patient as the project includes quite a number of sub-projects to tackle along the way.
The first design hurdle you need to decide is what technology level do you wish to do this at? With or without a microcontroller (a largely self-sufficient microprocessor), discrete logic (e.g. AND, OR, NOR gates and flip-flops) with/without arithmetic units (ALU), programmable logic (CPLD, FPGA), something else I haven't mentioned or considered. This should be first about the technology used to do the calculations, the input/output controls are secondary decisions (LED seven-segment displays, LCD panel) mostly influenced by aesthetics or cost.
One potentially useful starting places for learning about digital computation is the very accessible book, How Computers Do MathHow Computers Do Math (ISBN: 0471732788) by the quirky Clive Maxfield. This is written at the "soft" - programming or logical level, which you will need to understand in order to actually do the calculations.
Someone else mentioned the uWatch (-- micro-Watch) project as an example, and there are references around the Internet to electrical engineers (or EE students) that built their own calculator in the 1970s. Or this TI-59 clone (MS-Word doc) based on a dsPIC30 microcontroller. There are also some details on building a FPGA (programmable logic device) based calculator.
For a total novice to electronics (or digital electronics) I would suggest using a micrcontroller as a starting point in your design, look over the mentioned book's web site for a feel of the complexity in programming (not much if you have any programming experience) for the microcontroller and go from there.