Just take it one step at a time...
The LCD & touchscreen are probably the most difficult parts.
I'd break-up the project into the following parts:
I usually like to start with the easy part, so I'd probably start with the temperature sensor. Then add the Peltier on/off control, and then I'd hard-program the Peltier to hold some preset temperature with no user interface and no display.
If you want to simplify it a bit more, you can start with an LED (possibly the Pin-13 LED already on the board) in place of the Peltier and then heat & cool the temperature with a hair dryer & ice to see if the LED goes on/off when the Peltier should be going on/off.
But the main thing I am lagging behind is the code.
Read-through the language reference to get familiar with what the Arduino can do (the LCD will require an additional library, but don't worry about that yet). You won't understand or remember everything, but it won't take long to read-through the entire language reference.
Then, look through some of the examples and try some of the examples. Try to understand what the code is doing.
The two most important concepts in programming are conditional execution (if-statements, etc.) and looping (doing things over-and-over, usually until some condition is reached.
Once you understand if-statements and loops, you should be able to begin structuring your program.
When you write your program, write, compile and test (and debug if necessary) one or two lines of code at a time. The biggest mistake beginning programmers make is to write the whole program at once.... Then, the complier reports hundreds of meaningless error messages and you're stuck...
This isn't as easy as starting at the top of your program and working down... i.e. If you delete the bottom half of a program, it won't compile... You have to "develop" your code in a way that makes sense to the compiler.
As you develop your code, make use of the serial monitor so you can "see" what your program is doing. You can send messages such as "Turning-on Peltier", or you can send variable values or raw ADC readings from your temperature sensor and the resulting temperature calculations, etc.
Once your LCD screen is working reliably, you can also send test/debug information to the LCD instead of to your computer screen via the serial monitor. You might even be able to create a "debug" mode that you enter via your touchscreen!