Are my gears setup backwards? I have a trike with Shimano shifters. There are 3 gears in the front and 8 gears in the back. Gears are changed by rotating wheels found on each handlebar. The gear shifters have a window telling me which gear it is set to.

When the chain is on the largest-sized gear in the front, the sifter says "3". When it is on the smallest-sized gear, it says "1".

But the opposite is true of the back gears. When the chain is on the largest gear in the back, the shifter says "1". And when it is on the smallest gear in the back, it says, "8".

Are my gears setup the wrong way?
No, it's set up the right way.  (In fact, because of the way the front and rear gear systems work, it's essentially impossible to set them up backwards: you'd have to re-engineer the whole system.)

Think about it in terms of how many times the wheels turn for each turn of the pedals. When you change to a larger-sized cog at the front, the chain moves farther for each turn of the pedals, so the rear wheel turns more times.  When you change to a smaller-sized cog at the back, moving the chain the same distance causes the wheels to turn a greater number of times.  So, in both cases, moving to a higher-numbered gear causes the wheels to rotate more times for each turn of the pedals.