Welcome to Civilization 5! Hope you enjoy the game.
I created an overview of the initial parts of the technology tree to provide some orientation. Unlike in Civ4 there is only one starting technology, Agriculture, meaning every civ starts with it and only it, allowing workers to construct farms.
The initial phase of the game sees a mix of technologies that allow your workers to improve your land accordingly and military ones for defense and/or offense. I will present those two branches separately. It is your choice when you plan on needing which combat units. Unless you have something good to do with those units, go for the non-military technologies first.
Tile Improvements
As a rule of thumb in the inital phase of the game you first want to hook resources that provide extra food (fish only receives a bonus on gold, not food with a fishing boat), then production, then happiness, then gold. Obviously those overlap often. In general you want to research the technologies you require to hook the resources in your empire quickly - delays regarding this should only happen for a good reason. A reason as rushing for a wonder (technology), getting Archery to harass an opponent early on etc.
- Animal Husbandry: Pasture, Horse reveiled
- Mining: Mines, Clearing Forest
- Pottery + Calender: Granary, Plantation
As for granaries - you will want them eventually, though in this very initial phase Pottery is likely only worth researching due to being on the road to Calender.
- Wheel: Road, Water Mill
- Trapping (requires Animal Husbandry): Camp, Trading Post
- Masonry: Quarry, Clear Marsh
- Sailing (requires Pottery): Fishing Boat
Initial Military
When you start getting units depends on what you want to achieve. If you want to rush, getting Archery or Wheel for archers and chariots quickly might be a way to go. If you just want to defend, Archery and Bronze Working relatively fast might be a way - you might even feel safe from attacks for whatever reason and not in attacking mood yourself and first get all the needed improvement techs before beginning to research for units.
- Archery: Archer
- Wheel: Chariot
- Horseback Riding: Horseman
- Bronze Working: Spearman
Tech Paths after Initial Buildup
After you´ve established your first cities and improved land and/or did some initial fighting already, you will have to decide which path you want to pursuit now. While as usual in civ you are not bound to following one of the three tech orders I´ll descrive exactly, they are focused at achieving a specific goal.
- Iron Working -> Metal Casting -> Steel, potentially Mathematics
- Allows: Swordsman -> Longswordsman, reveils Iron, potentially catapults
If you´re planning on attacking or being attacked by a near-by opponent at this point in the game, this can be the path to follow. It takes some time to get to Steel, which can be used to get cities going well and being ready - potentially having build a barracks for instance - once Swordsman and Longswordsman become available. If you feel like terrain and positioning towards your opponent allow for it, get catapults as well to support your Swords-/Longswordsman.
- Construction -> Currency -> Engineering
- Allows (among other): Lumbermill, Colloseum, Markets, Bridges over Rivers
If you decide on expanding and improving land further, basically starting a second expansion and "buildering" phase, you can go along this route. Especially if you have a good amount of Forests in your land, this can be rewarding since Lumbermills provide those with a significant production bonus, making them a production powerhouse in a Golden Age.
Colloseums assist in controlling the unhappiness resulting from further expansion and Markets are there to help paying for the buildings in all those cities and generally in keeping income positive as the game progresses.
- Writing -> Philosophy -> Civil Service
- Leads to: Chivalry (Knights)
- Allows: Library, Oracle, Irrigated Farms +1 food, Pikemen
Another possibility is teching towards Civil Service. This is a path that can boost your technology output and support early Great People making.
Libraries increase science for each population in the city they are build in. Increasing population works well with going for GPs since both require food surplus. Here the library helps as well allowing to work up to two Scientist Specialists. The additional food from irrigated farms with Civil Service again helps in boosting science and/or working more Specialists.
This path eventually leads to Chivalry - it can be combined with Path 2, following a heavy early expansion phase towards five to six cities. Ignoring Path 1 and its technologies and instead going for a combination of Path 2 and Path 3 will result in a very good buildup and lots of attack possibilities from Medieval on with Knights for example. This works very well in SP Civ5 - we´ll see how much a human will want to and be able to disrupt here.
and
Libraries increase science for each population in the city they are build in.
are the only two I noticed.