On tiny it's hard to have one step-by-step strategy you can always follow. I have sort of key points and I improvise in between.
Ability picks: always luck, plus some combination of PQ, economy, morale and diplomacy bonuses. I tried various combinations of them and didn't notice big difference.
Expansion: don't lease anything, your income is not going to be large (especially on rare) and payments would take a big chunk out of it. Playing on rare pay attention to the evil/good choices, otherwise worthless evil choices might be very valuable on rare. Something like +40% production or +45% starships on a planet make a big difference if you only have 2 (or 3) of them [planets].
Research to diplomacy, change government, build or buy (if still have money) diplo translators. Trade for necessary techs. I usually have to sell diplo translators to one or 2 of the races, I pick ones who are going to be weak. Primary techs to get are trade, deflectors and shields and if there're minor civs nearby impulse drive. Stop your own research.
Build freighters and establish trade routes, at the same time build few defenders.
If there's a minor race neraby, conquer it.
If finances are bad, Build banking centers right away.
With rare planets I develop planet infrastructure, with abundant I rather build starbases in my sector.
Depending on the situation I often have to interrupt development to build few battle axes.
Then I try to either get some trade good, so that I can exchange it for the next generation of techs. If I can't build trade good, I will built more battle axes and trade them for tech. Get tech for battle cruisers early, this ship is [only] useful for trading.
AI is not willing to trade military techs, so I usually research them myself (starting from Battleships or sometimes from Interstellar Tactics). I don't bother with corvettes. Usually AI gets battleships first, but it's not a problem by itself. They took their time to build them. In some cases I may buy battleships though it's very expensive. To reduce the cost I may trade all my trade goods, techs + bunch of ships for it, however I will avoid paying high monthly payments, because it cripples economy too much.
Up to that point I don't worry much about influense or resources. Though I do my best to avoid wars at this stage, sometimes somebody (usually Drengin) will declare the war anyway. I just defend my home sector by battle axes and by building starbase improvements. Eventually AI gets tired of losing ships and makes peace, which give an opportunity to ask him for a few techs:). These wars is one of the reasons I don't go much for resources - it's too hard to defend them outside the home sector.
But this changes after I get dreadnoughts, which I will do quickly after getting battleships. I start actively build battleships while researching dreads, so that I can quickly switch production.
The goal of the next phase is to build up a fleet of few dreads and comabt transports and have couple constructors in reserve. I may not complete this stage if there's good opportunity to join the [war] party

If 2 races are fighting and see easy planet to pick (preferrably somebody's last) I go for it even if half ready. If such opportunity doesn't present itself I finish the phase and go to full scale war. When fleets are in position, first step is to take out military bases.
How to pick up an opponent for the first real war. This depends on many factors:
- Drengin often leaves no choice but go against them
- somebody have 1 or 2 nice planets left. Remember, you'll inherit their various "capitals"
- somebody has good resources (economy,morale,military)
The first war must be finished quickly, so I don't start it until I have enough force to complete it in 1 or 2 months.
This actually concludes the difficult part of the game. The next step is inevitably to built up some more power, let the population grow a bit, buy few weapons/defense improving techs (AI doesn't charge much for them) and go to war again. This time I usually go against the strongest race, because they're still developing/growing faster than me. Funny thing at this stage is that sometimes I just don't have enough population to take them over, so for a while I have to patrol the space and destroy their ships as soon as they build them:) This may last for a few years until my population will grow enough:)
After that you can choose any victory method you like.