Choosing a good city site is critical to getting a lot of Trade Arrows as your city grows. Look especially for Trade Special terrain tiles with things like Wine or Gems or Gold. These have 4-6 Trade Arrows each, although their production or food harvests are relatively low. Early on you want your city to grow quickly, at least as quickly as you can provide happiness improvements to keep the citizens content. Working a lot of Forest or Mined Hills will give you good production, but your growth will be stunted due to lack of excess food and your research will be stunted due to lack of Trade Arrows.
The other way to get piles of research beakers is by using caravans to get large trade bonuses. This happens after you discover Trade, which most people put off until after making their first government upgrade. You can use caravans to move shields from one city to another for Wonder-building, too, so critical decisions need to be made about use of early caravans. The bonus payout for delivering a caravan to another civ far away is much higher than delivering to one of your own cities, but it takes longer and you lose one end of the Trade Route benefit. Once your trade system gets going you can ratchet down your Science level and rely on caravan deliveries for your research. Of course, about then you will be needing to raise Luxuries to keep bigger cities happy, and Taxes to finance building improvements and "neighbor resettlement" programs. There is no single answer for anything in CIV, you need to learn how the different elements interplay, but Trade is the most often neglected aspect of CIV.