If I connect 2 charged sides of battery by copper wire and I touch
that wire will electricity travel through me?
Yes, it will.
If the copper wire is carrying some current it means that some voltage is dropping along it, i.e. if you measure the voltage at one point it will differ from the voltage measured on another point. When you touch the wire you are touching it in many points, so that your finger would experience different voltages in different points, hence a current. Please note that this current would be very, very very weak for most practical cases, so the average engineer would say "no current at all, man", and he'd be right.
There also is some current that would flow from your finger to ground through your feet, but again this current would be negligible since battery voltages usually are low and the human body resistance is high enough to keep this current low (and safe). I'd say that 50V is the turning point: if you touch a wire above that you might be electrocuted and possibly suffer some injuries.
If the electricity in electric lines doesn't travel to ground(grounded
circuit) in normal circumstances, why does it shock me when I touch
the line(I thought electricity travels only through path with less
If I understand correctly the question is: "since electric lines are not connected to ground, i.e. current flows from the generator to the load through the hot wire, and back through the neutral wire, why should I be shocked touching only one wire?"
That's because ground is also part of the circuit, and also if it wasn't it becomes part of the circuit when you touch the hot wire with your hand, and the ground with your feet. The neutral wire is connected to ground in the transformer cabinets for safety reasons, so that touching the neutral should not be harmful and wire insulation may be properly sized. Think of a floating line, with neutral at some kV above ground and hot some 230V (110V) above neutral. That would be very, very bad. When you touch the hot wire it's just like you are touching also the neutral, thus you get electrocuted.
I'd add a few lines about that "I thought electricity travels only through path with less resistance". This is false. Given a fixed amount of current, most of it will travel through the path with less resistance, but some of it will travel where resistance is higher. Think of resistance as the width of a road: the wider the road (less resistance) the higher the number of cars that can pass through there. In narrower roads you still find cars, just a smaller amount.
About the circuit you sketched: no current is flowing into ground in that case. Ground is only providing a reference and makes sure that a user, that is connected to ground too, won't get electrocuted touching the wire coming out of the '-' terminal of V1. Moreover touching '+' terminal of V1 should get you electrocuted less, since V1+ is only 1V above ground. If you disconnect the ground the '-' can go for example at 1kV, while '+' should be at 1001V. If you touch either wire now you get some serious current through your body.