Can you simply increase the voltage going to the antenna, or is it more complicated than that?
An antenna doesn't have range. A transceiver system has a range of operation.
Often, that range is limited by the minimum SNR (Signal-to-Noise) that the receiver needs to operate successfully. In that case: Increasing the transmit power (which is what increasing signal voltages does) will increase the receive power, which will in turn increase the SNR (assuming noise stays constant). And then, you can move your receiver further away from the transmitter.
For many systems, it's not quite as simple. For example, in cellular mobile phone systems, the cell size (i.e. the radius of the circle around in which phones use a base station) is of course limited by the transmit power. But: The operators are actively keeping that power very limited, because it will lead to interference in other cells, which looks like Noise to the phone (and that's the best case). So, if a network operator simply increased all base station's transmit power, the coverage (and thus, the range) would actually become worse, not better.
Also, unless you are in space, when your signal travels farther, then it will also show more effects of non-ideal channels: For example, consider the multi-path channel, where a signal doesn't only have a single way of reaching the receiver, but maybe different paths with reflections in between (that is very typical for urban microwave scenarios, i.e. mobile phones, WiFi, the likes of that, but also DVB-T and so on). Now, the path length difference between the shortest and the longest path (we call the resulting time difference "delay spread) has a direct influence on the channel bandwidth that you can use (until you need very capable equalizers). Now, if your power is limited, your path stays short, and weak reflections just don't hurt as much. With more power, you get more range, but then you get a potentially greater delay spread, which means your system must resort to using a shorter bandwidth, which in turn might reduce the amount of error correction you can do – and you end up with less effective range.
Also, "simply" increasing the voltage to an antenna usually breaks laws, makes amplifiers harder and expensive to build, things more power-hungry, and will also introduce non-linearities (and that means bad distortion) in systems if they're driven into nonlinearity: A sensitive receiver next to a very "loud" transmitter will actually get worse reception, not better.
NASA uses 200 foot dishes in Madrid, Canberra and Goldstone, in the DeepSpace network. Despite these parabolic dishes, and liquid-helium-cooled maser-amplifiers up in the dish, datarates from Pluto were just a few hundred bits per second.
Normal antennas are often single quarter-wave ( 1 inch for WIFI at 2.4MHz) pieces of wire. Adding one slightly longer piece of wire(call it the Reflector), properly spaced, along one side of the original wire (called the Radiator) begins to shape the entire "beam" away from a uniform pattern to a more main-lobe response.
Adding shorter pieces of wire along the opposite side (call these pieces the Directors) will further shape the beam. I've seen these antennas, called Yagi, with 10 or 20 Directors. Look at the TV antennas for fringe/rural areas.
Does this addition of tuned pieces of metal increase the "range"? If range is the distance between transmitter and receiver, with successful signal reception, then yes.
Other mechanical/electrical structures for antennas exist. log-period-V is one.