I'm trying to make an electromagnet that's strength is constantly getting incremented by small amounts every second. I need to know, which would have a greater effect on the electromagnet's strength, amps or volts? (I know increasing the turns and/or density of the magnet wire will increase the strength, but I am looking for answers other than that particular one.)
Firstly, voltage plays no part in the strength of an electromagnet, it's only the current through the windings that generate the field. Consider a super-conducting magnet with zero resistance windings. There's no voltage, no power dissipation, and a large magnetic field.
However, most of us don't have the luxury of using a super-conductor to wind our magnets, we have to make do with a good-conductor, like copper. As it has resistance, we need to apply a voltage across it to push a current through it. This results in bad things like heating in the coil, whose temperature needs to be limited if it's to keep working.
For any given wound magnet, we do not have independent control over voltage and current. The magnet coil defines the ratio of those, it's called its resistance. If we apply a voltage, then (voltage/resistance) current flows. If we connect a current source, then the voltage across the coil responds, and becomes (resistance * current).
The limit for both voltage and current is given by the coil's cooling. If we operate the coil for a very short time, we can increase the power supply, and allow the heat to be stored as a temperature rise in the windings, as long as we switch off before the coil overheats, to let it cool down before the next pulse.
In reality - it's both. The strength depends on the current, and the current depends on the DIFFERENCE between the applied voltage (e.g. the battery) and the INDUCED voltage (how fast the electromagnet is moving in relation to whatever strength you mean - usually a permanent magnet).
If you move or spin an electromagnet within a magnetic field, that causes (induces) voltage to flow through the coil. The faster you spin/move, the higher the induced voltage. If you spin it fast enough, that induced voltage becomes equal to the battery voltage (which means no current can flow at all). If you spin it faster still, it generates even more voltage than the battery, and you end up with a negative strength (charges the battery - otherwise known as regenerative breaking).
Most motors using electromagnets carry a "KV" rating, which tells you the speed it needs to be going to induce 1 volt (or the opposite - multiply it by your battery voltage to work out the fastest speed it can turn).
So to answer your question - you need to know how fast things are moving - if it's slow, a low voltage is fine. If it's very fast, you'll need a high voltage. After that - the current changes the strength, OR, the amount of time you turn it on for changes the strength. Normally, you cannot easily "change the current" - instead - you use PWM switching to turn it on and off at full current for short amounts of time...