I have a 2014 Opus with Shimano Tiagra 10 speed. I have seen Tiagra shifters which don't have this annoying cabling coming out of the shifter and instead, they are together with the brake cables which goes underneath the handlebar grip tape. Can anyone suggest me how to upgrade the necessary components of the shifters to get rid of these cables in the front? I presume I might be able to save some watts on higher speeds as these flaps a lot.
Newer versions of Tiagra brifters route the shift cable under the bar tape. Do be careful though because Shimano has changed their cable pull on some versions. The 4700 series Tiagra uses a cable pull ratio that is different from any other Shimano models. If you get 4700 brifters you would need to replace your derailleurs to be compatible.
you will have to guide the cable underneath the bar tape. You could re-use your old bar tape if it's still in good shape.
You might need to change the length of the cable housing but judging from the image you should be fine with the current length.
I usually tape the cable housing to the handlebars in order to keep it in place whilst I wrap the bar tape (using some electrical tape). Some handlebars have a built in cable groove, be sure to use it if it is present.
You will indeed have to upgrade to a newer style shifter which does not have the shifters cables exiting from the side, some suggestions for this can be found in the other answers. Modifying your current shifters to route the cable to the back is not very viable in my opinion. This video shows you how to install/guide the cables for the new shifters: