I'm trying to trim a part of a song in Audacity and merge the other two parts together, but right where I cut it you can hear a sudden difference in the tempo and audio. Is there a method or some sort of software that could make the transition from one part to the other sound more "smooth"? (There's not a crazy difference in the music between these two parts, I really do think it's possible to smooth the transition, if somehow possible).
Yes there is. It's called a crossfade and is a very basic part of audio editing technique.
As a crossfade starts, you are playing two clips at the same time, however for the duration of the crossfade you are reducing the gain of one and increasing the gain of the other. At the point where the first clip is no longer audible, the second clip is at full gain.
The 'shape' of the fade can be programmed in various digital audio workstations. Common shapes are 'linear' and 'logarithmic' - allowing equal 'gain' and equal 'power' across the crossfade respectively.