A few things to try:
- The Palace grants 1 science and culture for each adjacent quarter. It doesn't sound like much but it does add up.
- Adjacency: Place science/culture buildings where they can have the highest possible adjacency (Science buildings from resources and wonders, Culture buildings from mountains, NW and wonders). Try to build Wonders next to these.
- Science/culture buildings: Build them as early as possible and not in the same Quarter.
- Science/cultural Independent Powers: They offer powerful bonuses so befriending them early is a priority. If you think you can suz lots of them, the +1 science or culture on the respective building per IP or % per IP will push your base rate up a lot. This will be tougher on a Small map though as there are less IP's.
- Attributes (and Mementos): Attributes are hugely important for raising yields. Both the Culture and Science attribute trees have relatively early nodes that can provide a really significant amount of their respective yields. Getting 3 of the respective attribute points can get you the +1 science/culture on resources in your settlements. +15/30% to yields in specialised Towns under the Expansionist tree can also be powerful as is the +3% per alliance to all yields under the Diplomatic tree.
@Lord Yanaek made a great guide to gaining attribute points here:
https://forums.civfanatics.com/threads/a-guide-to-attribute-points-how-to-get-more-of-them.698260/
You can also begin each age with a science and cultural attribute point memento to get you moving down the tree. Each future tech or civic that you complete in Antiquity gives you a Wildcard attribute point to spend.
Towns/Cities: Expand as quickly as possible. More Towns = more gold, resources and other yields. With Augustus you can buy Culture buildings in Towns at a 50% discount so absolutely do that before converting a Town to a City. You don't necessarily need to convert a Town to a City as early as possible. Convert them when you can place resources that provide production into them so they can build things faster. Or convert them when you have enough gold to buy a Science building straight away as well.
Golden Ages: The golden age academies or amphitheaters carry their yields to the Exploration Age so are a good source of the yield.
Tech tree: There are techs masteries that add science or culture to the respective buildings. Research them quickly!
- Specialists: In Exploration, I tend to start adding specialists when a science/culture district is available. In other words, it can be very early in the age. For example, once the Observatory is built in a City, I usually put the next available Specialist there and complete the Quarter with the first available Production building or University. (If I've taken a Scientific Golden Age, I'll often put the Observatory in with the GA academy so the Quarter is completed with a large science output.)
