Weight training for swimmers is essential to supplement the hours you spend in the pool. Your body gets used to do the same thing over and over again so your strength will not improve much by just swimming. Weight training is a great way to add some more strength.
Weight Training For Swimming
You need to be careful when doing weights for your arms alongside swimming that you don't bulk up too much and lose your bodies natural shape. Bulking up too much can make you less flexible and also less streamlined.
A good starting point is doing some push-ups as you are just using your body weight. But if you specifically want to do some weight training then a barbell bench press is a great starting point for doing some weight training for your upper body.
Weight Training For Swimmers Legs
Now generally the stroke that requires the most powerful legs is breaststroke. But powerful legs are essential for sprint swimming, not so much long distance as that requires more endurance. But you need lots of explosive power in your legs for sprint swimming.
Dumbbell lunges are a great weight training exercise for your legs. Here is an example video of how to do dumbbell lunges.
Plyometric exercises are also something you should be definitely looking into when trying to improve your explosive speed. Plyometrics build explosive speed and strength into your muscles which is perfect for swimming - so rather than just bulking up you are training your fast twitch muscle fibres to react quickly.
Some More Dry-land Workouts
Weights can get a bit boring so it's always good to mix up your routine with some different exercises. Tools like medicine balls have been used for decades by swim coaches to help build strength out the pool. Here is a video below of some of the swimmers at Indiana University doing some strength training.