I recently purchased my first bicycle since I was a kid. The bike I bought has individual gear levers for shifting up and down for both the front and rear gears. I thought it was odd that these levers are set up in such a way that shifting the lever with my thumb shifts up on the front and down on the rear gears.
Is this typical to all bikes with front and rear gear systems? Is there a way to change it?
Gear shifting has historical lineage, not always "logical" or intuitive by modern logic as observed by newcomers to the activity.
Originally, cables actuated the pulling of a mechanism, and release meant relaxing the tension, which mostly then let a spring return the mechanism to its lowest state.
This translated to pulling = lifting/pushing the system up, against the gears, so moving the chain from smaller, up onto bigger sets of cogs. Dropping = the easier task of letting the chain fall from big to smaller rings, which the internal spring could handle. When indexed shifting arrived about 30 years ago it just aligned the clicks to a particular gear; then, first in mountain bikes, then road, the integrated levers made one lever for up, the other for down.
The issue you're asking about relates to the fact that "bigger" gears up front do increase your gear, but in the rear they decrease, or lower your gear. The release of most thumb levers relaxes the cable, which drops the front to a lower gear, but drops the rear to a higher gear.
With a few unsuccessful exceptions that tried to use the spring to push the chain onto bigger sprockets, this is still the general pattern - EXCEPT Now, I understand Electronic systems can be programmed entirely different, even mixing left and right signals front to back, an exceedingly confusing idea in my opinion.
Learning to shift the traditional logic is easy enough to master in a couple weeks, so don't get worked up about something thousands have managed to cope with just fine.
Grip shifts are my preferred MTB choice, but again the tension/detension concept applies relative to which way you rotate the barrel.
This is normal for Shimano trigger shifters, and you'll get used to it quickly. SRAM trigger shifters can be operated up and down with thumbs. And there are grip shifts which turn a bit on the handlebars.
Aside from possibly trigger->gripshifts, you'd be doing a decent amount of drivetrain overhaul.
Mechanically, the shifting (front & rear) are the same. Moving either shifter so that cable is pulled, causes the chain to move onto a larger sprocket (front chain-ring or rear cog) & releasing cable allows spring tension in the derailleur to move the chain to a smaller sprocket. However, a larger chain-ring results in a larger gear ratio (higher gear) and a larger rear cog results in a smaller gear ratio (lower gear). So, the same action on front or rear shifter has the opposite effect on gear ratio. In practice, it only takes a little time before you begin shifting automatically; you will shift up and down on the front or rear without thinking about it.
