Having done it before as well, I would suggest three trips. Doing it this way, you'll have enough space money to buy each ship after the previous trip.
Trip 1 - something "short" just outside the bubble. probably about 500-1000 light years. Do this in your current ship. You'll need the best FSD you can get, a fuel scoop, and (ideally) the advanced discovery scanner. This will be mostly a learning trip, and should fund enough to get an exploration cobra.
Trip 2 - somewhere like the coal sack nebula (that was my first trip!). about 2,000 to 3,000 outside the bubble. Now you'll start getting your name on undiscovered things. doing this will get you enough (or close to enough) for a diamond back explorer (or if you get really lucky, an Asp explorer).
Trip 3 - to Sag A*. You'll know enough by now to be fine. watch out for neutrons!
Obviously, for all these builds you'll want the best FSD, fuel scoop, and scanner(s) you can get. keep it as light as possible. read the exploration write-ups that are all over. several really good ones. Have fun!!! explore for fun, not for money.