Because I apparently can't plug the Uni-Wiki enough, their page on Research explains most of what you are looking for:
- Laboratory Operation I
- Research I (5% bonus to blueprint manufacturing time research per skill level)
- Science 1 (5% Bonus to blueprint copying speed per level)
- Metallurgy 1 (5% Bonus to material efficiency research speed per skill level)
- you should have Production Efficiency trained to level 5 before you even start producing.
- Advanced Laboratory Operation (+1 lab slot per level)
- Cybernetics 5 (allows to plug in special implants that speeds up the research process)
- The Beancounter Hardwiring series
You can reduce this waste by researching the Material Level (ME) of the BPO. You can see the design waste on the BPO under Wastage Factor. This is always 10% (0.1) except for some rare items that have a 5% (0.05) Wastage Factor. Drones used to have a 5% wastage factor, but apparently they don't anymore. By researching the BPO we can reduce the design waste using the following formula: 1 + Base Wastage / (1 + ME level).
Using the above formula on a BPO with 10% base wastage, we can calculate our production waste (assuming PE skill at level 5) to be:
ME0 - 10%
ME1 - 5%
ME2 - 3.3%
ME3 - 2.5%
ME10 - 0.9%
ME50 - 0.2%
As you can see, training your ME level from 0 to 10 reduces the waste from 10% to 0.91%, thus reducing the waste by over 9% in simply 10 levels, whereas training a further 40 levels (11-50) only takes off 0.7% (from 0.9% to 0.2%). The following graph can better illustrate the effect of training ME to higher levels.
A general rule of thumb used by the producers is that 30 days of research is usually sufficient.
The Productivity Level of a blueprint, as mentioned above, affects the manufacturing time for each batch made with the blueprint. Specifically, it reduces the manufacturing time by (Productivity Level) / ( (5 + 5 * Productivity Level) ) which results in the following progression:
PL 1: 1/10 reduction in time
PL 2: 2/15 reduction
PL 3: 3/20 reduction
PL 4: 4/25 reduction
PL 5: 5/30 reduction
...and so on.
For example, suppose that manufacturing a batch of something normally takes 60 minutes to manufacture at Productivity Level 0. At Productivity Level 5, this is reduced by 5/30, or 10 minutes, thus the manufacturing time for one batch is 50 minutes.
Increasing the Productivity Level of a blueprint is usually not as important as increasing the Material Level of a blueprint. It might be important in cases where the blueprint is being used to manufacture almost continuously or in cases where the manufacturing time for a single batch is extremely long.