What equipment do I need to complete a SR series?
I'm not going to go into too much detail for this as it's a very deep and personal subject. I think the primary thing is not to get too hung up on not having quite the right piece of equipment or bike. Don't let that stop you (within reason. If you only own summer clothing, then you will have to invest in some warmer clothing if you want to start an SR series in winter!). Almost any bike, so long as it is mechanically sound and fits you reasonably well, can and has been used to ride an SR series.
Rushing out and spending huge sums of money on a new bike is not always the best plan. Ride what you've got and as you get the experience and talk with other riders, work out what you want out of a bike. You will see anything and everything being ridden, from the super-exotic carbon machinery to things that look like they were rescued from a skip (and probably should have remained there!).
In terms of bikes, well, my opinion is that comfort just about trumps everything. You're going to sit on the thing for 20-30 hours on a 600km ride, it needs to be comfortable and not beat you up too much. How that comfort is achieved is up to you. In the days of old, steel was real and seen as the comfy ride. These days, carbon bikes with big tyre clearances are becoming a more and more common choice for riders. The key thing there being the tyres. 25 and 28mm tyres are more typically used by randonneurs as they offer much greater comfort than 120psi 23mm tyres of old! We were often sneered at by 'serious' road riders with their thin, super high pressure tyres that must be the fastest type possible. Well, turns out, wide tyres are faster and more comfortable. Win.
Clothing and equipment - again I won't go into too much detail here. Partly as with clothing we'd have to look at the different seasons in detail. The key is working out what works for you. A lot of people look at the amount of clothing I wear and have in my bag - I use it all! I feel the cold badly, so carry more layers than most. But, there are those who are happy in shorts and a short-sleeved jersey in the snow. Everyone's different with clothing!
Equipment - a minimum you need a spare tube (preferably 2), tyre levers, pump and a small multi-tool. That's it. Plenty of people ride with just that (no-one should ride with less and you will not get a good reception if you're broken down and do not have these basics). Beyond that, again, it's a question of working it out, however, I think that my post on packing for PBP covered much of the possibilities (however, others carry more or less than me for sure!).
The one thing the above did not go into in detail is lighting. Lighting will be essential for riding a long way especially if you start your SR in the traditional depths of winter. You need something good, a little blinky to be seen by will not cut it in the middle of the night on fast descents. You do not need an expensive dynamo hub and light to start with (though many will move to them over time as the convenience is hard to beat). There are plenty of decent battery powered lights out there now, you need to be confident that they will last through the night and/or have replaceable batteries (preferably something you stand a chance of buying on the road such as AA).
But, as with everything, work with what you've got and then over time figure out which direction you want to go in.
Finally, navigation. You do not need a GPS to ride a long way. They do make it easier in many ways, but people have been riding a long way using paper routesheets for many decades (though remember you need to be able to read it at night (which is the biggest issue for me) so you will need a headtorch or similar).