Listening to your piece, there's a number of fundamental issues that immediately jump out and should be given attention before you focus on transitioning/modulating from one key/phrase/section to the next. Although this is all quite subjective, there are certain conventional guidelines you can follow to get your music sounding better, or as you put it, more natural. As a starter, I'd take the time to investigate the following concepts.
- Time Signature - the underlying tambourine (and kick/snare?) appear to be painfully out of sync with the rest of the composition. For your music to sound 'together', there should ideally be a core pulse or rhythm that everything follows. Once you've grasped this, you can start syncopating various lines.
- Key Signature - although I'm not detecting any dissonance, it doesn't sound to my ears that you've explicitly chosen a key to work with, rather that you're just picking out notes that sound okay together. Similarly, focus on building a solid chord progression and stick to it.
- Busyness - it sounds like you've tried to cram so much into the composition that it just ends up a mess of competing sounds and melodies, i.e. spaghetti. Try and start small by composing and arranging a piece for a quartet for example, or perhaps with a piano, cello and violin which could provide a chord bed, a bottom line and a melody respectively.
- Lack of Dynamics - for a piece of music to sound interesting and varied (again subjective) it should ideally have moments of quietness to contrast and balance the moments of volume. This is generally referred to as dynamics. Try and focus on bringing some 'space' into your work instead of going hell for leather from the outset.
- Sound Quality - this is to do with both the quality of your sounds and the timbre and character of each individual component instrument. For more realistic sounds that are free try Googling 'Sonatina' and 'DSK Overture' (my low rep disallows too many links).
In a nutshell, slow yourself down and listen to lots of the kind of music you wish to make. Broadening your listening material is a very healthy thing to do and helps to train your ear. You can learn a great deal from simply listening and dissecting other works.
Hope this helps, good luck.