1. Before you start researching, you should think about the route you want to follow- researching without giving it a second thought will cost you a lot of precious time and can even lead to a scenario in which your units will be unable to keep up with enemy soldiers.
2. There are three routes available from the beginning of the game, which will give you different profits upon completion - unlocking new weapons, the ability to equip your soldiers with better armors, or unlocking the Psi Lab facility, which will give you access to the new and extremely powerful Psi Operative soldier class. The last option is the more profitable in the long run, as the Psi Operative class requires an immense amount of time for training.
3. You shouldn't focus on a single technology without thinking about its costs - a technology taking 40 days to complete, to give you access to an armor that you won't be able to afford either way is a huge waste of time. Instead, you can complete a number of smaller projects and focus on the bigger, more profitable ones when you have additional scientists.
4. Remember that in the case of soldier equipment (primary and secondary weapons, armors) you have to craft only a single "piece" of equipment, after which all of your soldiers will be able to equip it, regardless of the number of soldiers you currently have. It looks differently in case of the experimental and most utility items, as you can only equip them to a single soldier.
5. The influence of scientists on the research speed suffers greatly from the process known as "diminishing returns". It means that each additional scientist you recruit will decrease the time needed to complete the project by a smaller amount than the scientists that came before him - the first scientist you acquire will decrease the time by 30-40%, whereas the 4th or 5th decreases it only by 1/3 of that. You should aim to acquire about 2 or 3 scientists as quickly as possible and then focus on something else, as their impact quickly decreases.
6. Two research projects are absolutely mandatory and should be completed as quickly as possible. Those are the "Muton Autopsy", which gives your soldiers an upgraded version of grenades, as well as the "Faceless Autopsy" allowing you to craft an extremely powerful item called the Mimic Beacon.
7. Sometimes during hacking you will have the option to decrease the time needed to complete the research by 50%. If you don't have any other option you can take it, but other than that it's completely useless and not worth the risk. It's because of the fact that it decreases the REMAINING time, not the whole time of the research, which, in extreme cases, can decrease it by only a single day.
8. Black Market sometimes offer the option to decrease the time needed to complete various projects - those require a huge amount of Intel to be bought, but sometimes will allow you to save up to 10 days of research.