So what happens to your personal time and space, exactly, when you get close, or even match, the speed of light? 1g seems to be the value of choice. We get on the ship with sufficient fuel to provide 1g acceleration for a century. Off we go, aiming for some distant star or galaxy. Let's call this star Jeff. Let's further assume that Jeff is exactly 100 light-years away from Earth.
We clear the Solar System in less than a month and continue accelerating. At this rate we will reach the proverbial 0.97c in just under two years. It'll take us another ten years to achieve 0.99c, but let's leave that alone for now.
The space around us will shrink, making destinations more accessible. Our personal time will slow down to near-zero. Does this mean that in our personal time we will be able to reach Jeff in a matter of minutes? Or am I missing something?
The second part is a bit harder, I think. If by dint of scientific trickery we did, in fact, reach the speed of light, would our personal time slow down to zero, and travel between any two points in the Universe become INSTANT so far as we were concerned?
 A: Theoretically, yes. If you sped up enough then your personal time frame would be such that you could travel between 2 very distant points, between galaxies or farther, in a very short space of time. Of course that means not slowing down, as that would take years. 
The faster you go the faster time would be going by outside of your frame of reference. This raises some considerations:


*

*If you are going to fly by the Andromeda galaxy (just as an example) then you will travel 2.5 million light years, which means that about 2.5 million years will have elapsed outside your frame of reference no matter how quickly it goes by in your frame of reference. You would need to account for 2.5 million years of drift - galaxies are not stationary - and point where it's going to be 

*If you are going that quickly you will be passing Andromeda in a fraction of a second by your frame of reference, doing any science as you go by will be very problematic because of the distortions, just taking a picture as you go by could be impossible 

A: Physicist John Baez has a nice web page on this topic, complete with equations and sample data. Here is a table of times and distances that he gives:
    T               t              d           v                   γ
    1 year       1.19 yrs     0.56 lyrs      0.77c                   1.58  
    2            3.75         2.90           0.97                    3.99
    5           83.7         82.7            0.99993                86.2
    8        1,840        1,839              0.9999998           1,895
   12      113,243      113,242              0.99999999996     116,641

The first column is the amount of time as measured on your own clock, aboard your starship. The third column is the distance you travel. The third line is essentially the answer to this question. It will take you just a tad over five years (actually 5.18 years) to travel 100 light years.
A couple of other answers (one of them now deleted) have greatly overstated the extent of the time dilation effect near the end of the trip. Here are some actual figures for the time near the end of a 100 ly trip:
To travel the final 60 light years takes 320 days.
To travel the final 20 light years takes 65 days.
To travel the final 10 light years takes 37 days.
To travel the final 1 light year takes 3.5 days.


The second part is a bit harder, I think. If by dint of scientific trickery we did, in fact, reach the speed of light, would our personal time slow down to zero, and travel between any two points in the Universe become INSTANT so far as we were concerned?

This question doesn't have a scientific answer. Our current theories of physics say that you can't reach $c$, so if you assume we can reach $c$, you're assuming that those theories are false, and therefore we can't use them to get an answer to your question.
Rory Alsop complains in a comment:

Unfortunately your first paragraph is wrong. If your craft can accelerate at 1g from rest, it will not do that as it nears c. The craft's effective mass will dramatically increase - so you'd need to be able to improve your thrust by an amount approaching infinity as you get closer to c.

This is not quite right. Relativistic dynamics does make it extremely difficult to build a rocket that can do this kind of thing, but not exactly for the reasons stated. The 1 g is 1 g of proper acceleration, meaning the acceleration measured by an accelerometer on board the ship. It's the acceleration you feel. In the frame of an observer aboard the ship, there is no change in the ship's apparent inertia. As measured by the voyager, the same x newtons of thrust will always produce the same acceleration of 1 g. However, it is certainly not practical to achieve this sort of motion with a literal rocket. The amount of fuel required grows exponentially, and that's just a losing game.
There are other realistic problems as well, such as the fact that your ship would be destroyed by frictional heating as you plowed through the interstellar medium. The particles of the interstellar medium would also produce high-energy cascades of secondary radiation when they hit the ship, and this radiation would kill you. (Adding shielding doesn't work as well as you'd think, because it just tends to increase the amount of secondary radiation.)
The energy requirements for a literal rocket are also dire. By the time you've traveled 100 light-years, your $\gamma$ is 104. That means that the kinetic energy of your ship is $103mc^2$, where $m$ is its (rest) mass at the end. Let's just round that off to 100. So for example if your energy source was antimatter, you would have to have carried at least $50m$ of matter and $50m$ of antimatter. In fact it's much, much worse than that, because nearly all of the kinetic energy went into the reaction mass. The actual amount of KE depends on the exhaust velocity, but it's likely to be orders of magnitude greater.
