So many questions need to be asked...

Which race are you playing? What tech tree and which super ability?

Alot also depends on what you're building on planets. Are you doing mostly factories, research, or economy buildings? Are you specializing each planet or doing a generic build? Are you building immediately after colonizing or waiting for the population levels rise some? About what turn # are the enemies envading? (click the date at the top to switch it between date and turn #)

I normally play with a huge gigantic galaxy, Crippling difficulty, 9 Major / 8 Minor (aka cash cows), as a custom race with the Iconian tech tree and Super Breeder ability. Abilities are 30% Econ(4p), 20% Moral(3p), 20% Research(4p), 10% Diplomacy(2p), 25% Luck(1p) and 25% Creativity(1p). Politically I go with the Technologists for another 20% Research.

The first techs I race toward (after Universal Translator) are for the Precursor Library, Interstellar Refinery, and Molecular Fabricator. Initial Colony has 14 production, added to IR's 16 for 30, then MF's 50% bonus for 45 production with just 2 buildings. CivCapital is 24, +16 IR for 40, +50% MF for 60 production on the capital. I build the PL, IR, and MF on every planet. Normally after I get those 3 buildings researched, I'll focus on Diplomacy and Trade techs, with misc others mixed in as needed.

I can make the initial colony phase last for anywhere between 100 and 200 turns on a huge gigantic map, and about 80-150 on a large (run out of planets by then on large). Normally by the time the initial colony phase is over, sometimes while I'm still on the initial phase, I've started researching the hostile planet techs, and start colonizing those too.

During the colony phase, I don't build anything on the planets until they reach their population cap, except a starport. They're all set focus to research until then. Once they're at 8 pop, I'll build the IR, MR, and PL, then 1-2 moral buildings and fill the rest of the slots with economy buildings. At this phase I only do either Econ or Research colonies. With about 80 colonies there will be 10 research focus and 70 econ focus. It's not until the other races start using weapons that I'll convert some colonies into production so I can make ships faster. I can always bribe a race not to attack me, so I'm not worried about building defenses til my economy stabilizes.

edit: Started a new game today, and seems I was playing on gigantic maps instead of huge.

This will have some overlap with the previous posters, but I would see 3 possible reasons:

1) Tech trading is on and the AI is actively exchanging technologies. If tech trading is on, you DO need to use it or you will fall behind. Let us know if you want some specific pointers on tech trading (what to do and what not to do). Make sure you have a few research alliances with AI's.

2) Your building strategy doesn't focus enough on research. As Kontana already hinted at, specializing some planets is almost always a good idea. For example dedicate one of your higher quality planets completely to research (though start by building 1-2 factories so your labs can be built quickly; later you can still replace those factories by labs if you wish). The main reason for specializing planets is that capitals (econ, manufacturing, tech) give a bonus to the planet they are built on, so to get the most value out of your capital, build it on a planet that is mostly dedicated to that specific purpose. The only exception is the econ capital: it's often best to build it on your homeworld since that one has the highest population early in the game, so it will yield the most taxes.

3) Your research priorities are not optimal. I would suggest going for some of the cheaper technologies first.

- If you have tech trading on, don't ignore the diplomacy techs: higher diplomacy will get you better deals when buying, selling or exchanging technologies with AI's. In addition, diplomacy improves your relations with the AI's, which helps to reduce the chances of being attacked.

- Get to the tech capital fast: the sooner you get a boost to your research, the better you can keep up.

- This also applies to relatively cheap technologies that give a social / military production or economy bonus: getting these fast works exponentially: faster production allows you to build faster, which in turn gets you any bonus buildings faster, etc.

- Be quick to research one of the military branches (beam, mass driver, missiles), get bigger hull sizes and some miniaturization. In the beginning I often made the mistake of having a flourishing economy but not enough military techs to build a decent fleet quickly. 

Also, keep an eye on your relations with the AI: choose which ones you want to be friends with (typically it's easy to choose races with a similar ethic as yours: evil/good), set up 2 trade routes with some of these AI's as this helps to improve your relations; after 1 year establish economic and research alliances with them (preferably with as many races as possible). Try to pick some strong races as allies: first of all this reduces the chances of these strong ones attacking you, secondly you'll get the most benefit out of economic and research alliances if they are doing well.

Building on Bentley241's point: as soon as you see an AI that has a military rating above zero, build a few ships to keep up (or preferably stay ahead in military rating). AI's are far less eager to attack if your military rating is higher than theirs. It will also reduce the chances of being extorted by the AI for technology or money.

Feel free to bribe some of your future allies into attacking other AI's. If there is already a war going on, chances are less that anyone declares war on you + it weakens most of the parties that are at war.

As always, feel free to ask more questions...