The Lomo tank is a great start....it's what I have for super 8 and regular 8. Ensure the tank is clean, I mean really clean, and dry before you use it. If it is the tank with rubber hoses in/out, ensure they are still water tight and run lots of water through them too. Dry everything thoroughly before use.
For Tri-X, the process isn't too difficult. You can do B&W neg with similar chemicals/times to regular 135 Tri-X if you really want. Reversal processing isn't difficult either....just more lengthy and a little more temperature dependent. E6 colour is certainly temperature dependant and what I have done is float the tank and chemical containers in the kitchen sink at the correct temperature for a while to warm everything up. And add a little hot water as necessary to this "warm water bath" if the temperature drops by more than 1C.
And I have achieved results as good as the pro labs. It is just rather time consuming. The reversal process takes 2.5 times as long as processing a standard B&W negative 135 roll...the drying much longer....but it's fun and certainly doable. E6, process time is about the same to be honest. Note the bleaches used in reversal processing may well be among the most hazardous chemicals that you ever use in your home, depending on what you can get hold of and how much chemistry you do in your kitchen
They're highly unlikely to kill you but do note any warnings. Consider wearing disposable gloves, especially if your skin is sensitive to any of the chemicals. Don't use any mixing jugs, cylinders or funnels for food after using the chemicals....don't wash your photo kit in the dish washer (due to the way they tend to recycle water)...and if you wash in the kitchen sink be sure to thoroughly clean it before you wash any dishes in there.
Personally I have never bothered with distilled water but other than some calcium carbonate (hard water) my local supply is very good. Your mileage may well vary in terms of quality of tap water...warmer countries often have far more chlorine than we have in the UK, for example....or might simply be less "clean".