Well given a choice I will tell you how I would go about it. I would take a somewhat commercial approach and write my own data acquisition routine.
You need some form of current transducer as a front end so may as well start there. I would start with
one of these or a like unit. Units like these are very common available from dozens of sources. Maybe in a range of 0 to 20 amps or 0 to 50 amps. Range is purely a function of what current(s) you expect to see. I would look for the 0 to 10 volt output. So what you would have is 0 to 20 or 50 amps becomes 0 to 10 volts. So now you have a nice already conditioned and scaled low voltage proportional to the actual current. You are looking at about $120 USD.
That done you will need a data acquisition device. No sense on getting overly fancy. Here I would look at
one of these units or if you want better accuracy than the $29 USD version then maybe
one of these units. The latter unit cost more at $100 USD but affords 12 bit versus 10 bit resolution. Both units have a 0 to 10 volt input range to agree with the sensor. This makes for easy coding. Both units have recording software. I use the latter unit all the time for projects like this. Additionally DATAQ who makes them also has a pretty good SDK (Software Development Kit) so if you wish you can roll your own software.
I am not a programmer, not even close!
However, I have managed a few data acquisition routines in VB. Using the above hardware it is easy (even I can do it) to write some software (language of your choosing) to acquire the data and dump it into a data base or something as simple as an Excel spread sheet. Your call on that note.
You mention current. You could also get a voltage transducer similar to the current transducer I linked to
something like this which runs about another $100 USD. That gives you a current out of 4 to 20 mA proportional to a 0 to 150 VAC input. You run that through a 500 Ohm resistor and get 2 to 10 volts = 0 to 150 VAC. The merit to adding voltage is you can plot line voltage and current as to the load(s) that come and go on the line.
Now I would be remiss (and yelled at by other forum members) if I did not suggest using a micro controller chip. You would need to program the uC and make a board for it as well as developing and designing signal conditioning if you did not use the type transducers I suggested.
There are dozens of ways to go about this depending on what you want to do. I merely have suggested how I might go about it.
Ron