We are confused where to register the patent? US or Europe?
Patents are territorial. If you want to prevent competitors from using your invention in the US, then you'll need a US patent. If you want to prevent competitors from using your invention in Europe, then you'll need a European patent. Similarly for any other country you're interested in.
The decision about where to obtain a patent is a complex one. It generally involves considering market potential, where your competitors are, and the cost of obtaining and maintaining the patent in each country. If you never expect to sell your product in the US, and you don't have any competitors there, it probably doesn't make sense to get a US patent.
Most businesses would aim for the US and Europe as a first stage, due to the size and wealth of the respective markets. Then, if the business wants a bit more protection, generally China and Japan as a second stage. But it very much depends on what your business plan is.
In terms of procedure, typically you would first apply in your home country (either with the European Patent Office or a national one, such as the UK Intellectual Property Office), then within the next year apply in other countries (or file a PCT, to get an extra 18 months to choose other countries).
Which one is easier and less expensive if we hire a lawyer?
Neither the US or Europe is particularly easier than the other these days. The requirements for patentability are reasonably similar. The cost of an attorney for each is pretty similar too.
Typically you would use an attorney in your country. They would act as a go-between for any foreign attorneys you might need. While this is more expensive than engaging all the attorneys yourself, it makes the whole process a lot easier to manage for an inexperienced inventor.
Should we register under electronic hardware or software? I heard Software patents are almost never granted.
You don't really register under a particular category. You (or preferably, your attorney) drafts claims to cover your invention. If your invention is in software, then you would have claims to software. If your invention is in hardware, you would have claims to hardware.
Software patents are grantable if the claimed invention has a technical effect (in Europe) or is non-abstract (in the US). This generally requires that there is an effect occuring outside the software being executed. So software to better use memory in the computer is fine (since it affects the computer itself), but software to automate an algorithm is not (since it doesn't really do anything other than produce some data).