> I've recently started to enjoy working out.

then On the answer of this question below, you will find useful info regarding how to improve your grip:

[Painful grip on deadlift and lat pulldown][4]

> After some time though,

you might find out that you enjoy more lifting heavier weights (some of us do)
then for the rows ([like one arm dumbbell rows for instance][1]) I would try [weight lifting straps][2], they will reduce the strength you need to put on your grip.


for the pushing exercises, like dumbbell presses,
I would try [liquid chalk][3], it helps with your grip.



  [1]: https://fitness.stackexchange.com/questions/31796/one-arm-dumbell-bent-over-row-is-it-a-complete-exercise-for-the-back
  [2]: https://www.google.co.uk/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#tbm=shop&q=weight%20lifting%20straps
  [3]: https://www.google.co.uk/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=liquid%20chalk%20bodybuilding
  [4]: https://fitness.stackexchange.com/questions/1141/painful-grip-on-deadlift-and-lat-pulldown?rq=1