Increasing core strength and strength endurance is one of the biggest things that you can do off the bike to improve your performance and reduce aches or pains that you may experience racing. Most world class athletes probably spend hours a week developing core strength during the off season and smart ones will continue to do maintenance work during the season. This video outlines some of what I have settled on as the basic structure of my core strength routine. I will gladly add quantity to this and extra exercises when I can, but even with a basic routine that you may do three times per week for 10-15 minutes, you can do a lot to help your cycling, running, or multi-sport efforts.
Just start with what you can do, whether that's 5 push-ups and 20s planks or more. Rest as needed to start. With time you'll be able to do more reps and longer duration planks with less recovery between and your stability, strength, and comfort riding and running should improve notably.
If you want to add an extra level of difficulty that I really like, then you could consider getting a suspension trainer system. You can get higher levels of muscle activation with the added instability of doing planks on an exercise ball, bosu ball, or suspension trainer. TRX is the most well-known brand of suspension trainer, but you can get a more economical setup pretty cheaply that work great. At home, all I have is a pull-up bar, suspension trainer for planks, a couple sets of dumbbells for dead-lifts and lunges, an exercise ball, and a homemade box for weighted step-ups.
As always, keep things simple to start and work with your body to progress at a rate that you can handle. Make things challenging enough that it's good training, but always keep things within the realm of what your body can handle so that you can avoid injury.