(I posted about money in your other thread, so won't talk about that here).
Pull up the menu, on the left hand top click on 'Auto Turn' so that it is 'Off'. This way the current turn won't end until you click on the next turn button.
If you are trying to move one ship several times but move another sounds like you are trying to move that ship more spaces than it can go, so the extra attempts get tacked on to the second ship. At the bottom part of your screen in the middle you can see how many moves a given ship has (and how many it has left).
As to what research to trade, that's not something with a set answer. You can trade none or you can trade any tech you have that a given AI doesn't have, or (what most do) somewhere in-between those two extremes. When I do decide to trade a given tech I make sure and trade it to every AI I'm in contact with as a rule. I don't give the AI's weapon tech of any type, any tech that improves diplomacy, or any tech that allows them to build a wonder/trade good that I want for myself. Try to avoid trading the AI for its weapon tech, the AI values this tech highly and usually asks such a high price for it that it's just not worth it. I normally choose starting picks that boost research so I usually trade tech for money instead of tech.
If you plan on trading a lot, make sure to get the trade good Diplomatic Translators. This gives a whopping 40% bonus to your diplomatic rating which makes trading much more profitable to you. Heck, it makes -all- diplomatic actions more profitable (or less costly) to you. It's one of my 'must have' unique projects (Diplomatic Translators, Gravity Accelerators, and Tri-Strontium Steel are the three trade goods I make a big effort to ensure I get).