I was initiated into an Invercargill lodge last May, and have these few bits of advice for you...
Think of 'tendering the application' as the last (and an administrative) part of becoming a Mason. If you've decided you want to join - great! That's probably the hardest part. The next important step (IMHO) is choosing the right lodge. Since there a few lodges in Invers, it might help you to make contact with each lodge and meet some Masons. As your proposer will have to personally vouch for you, you will end up getting to know some brethren of your chosen lodge before your ballot, regardless of the number of lodges you've investigated. So get out there, meet the men, and choose the lodge that feels right.
The tattoo choice is obviously yours alone. I will echo the prior comments and recommend that you wait until after you're satisfied (for lack of a better word) with Freemasonry. The world has formed many and various opinions of Freemasonry, ranging from great respect, to quiet suspicion, to outright hatred and condemnation. Whether you like it or not, the symbol will associate you with an organisation that is maligned by quite a few people and factions. Until you feel prepared to explain, promote, and (like it or not...) defend the organisation, it may be best to wait on the tattoo.
That said, I'm thrilled to have joined Freemasonry, and I eagerly look forward to my 3rd degree next month, during which I will be raised to the status of Master Mason. The whole experience has been awesome and inspiring. I feel both proud and humbled, and i know that I still have so much to learn. I recommend the fraternity whole-heartedly. So... get out there and meet some Invers Masons!
Best of Luck,