the bike should be wired in automotive grade 16 gauge wire. on a stock Triumphs all red wires are positive and they originally came as a positive ground (earth) bike. you can change your bike to a negative ground if you want. the regulator/rectifier is that thing with the yellow wires going into it and if it smoked it's likely bad now.
Going by your questions it sounds like you really need to use the search on this site and use Google to search for Triumph M/C wiring diagrams.
Going by your questions it sounds like you really need to use the search on this site and use Google to search for Triumph M/C wiring diagrams.