Well, you can control your scientific spending from the domestic advisor, spending more money researches faster. Libraries, universities and research labs add 50% to the research output of the city it's built in. Finally, in the city manager screen you can assign citizens as scientists, which also increases that city's research output.
Thus, the key to research is money. Money is generated by production - a size 12 city producing Wealth surrounded by mines will give you extras cash to spend on science. Additionally, the more cities you have, the more cash you generate - so your strategy of early expansion isn't necessarily a bad one.