What could be a good project to realize neural network in hardware (VLSI) for a beginner?
As a beginner it is very easy to have broad ambitions such as "i want to implement a neural net in hardware". You must carefully define what exactly do you mean by that. If you are referring to a feed forward net with backtracking you must realise there isnt really any need for a fancy dedicated hardware for the actual net.

A trained neural net can be thought of as a look up table, so a general purpose cpu is sufficient.
Also the neural net itself (conventional feedforward) is really not some glorious arrangement of brain like logic that needs specialised hardware. Rather it is simply an arrangement of many adders, multipliers and a threshold function (which is also consisting of adders, multipliers etc) with floating point support. So if you really want to build it in hardware, you need to implement the aforementioned mathematical functions much like an ALU would.

Now if you want to implement hardware that speeds up training by invoking parallelism, there is some scope here. Start with an FPGA, implement floating point units, threads etc. This is not a beginners project.

If the main goal of this project is to have fun with training a neural network and understanding it on a low level your best bet as a beginner is to do it in software avoiding mainstream libraries such as the famous Tensorflow. If you want to do it low level, try implementing a Neural Net in C++ (or any other language for that matter) and train it using a graphic card using CUDAs API (Nvidia only) to recognise human handwriting from the MNIST database. It is low level enough to have real fun.