While MJeffryes has provided a good map to help in the general case, I'd like to answer your specific case.
There isn't really a good route entirely by rail or tube. However, I can spot a few decent-looking routes if you're willing to go for a longish walk or a short-to-medium cycle (I'm not a cyclist myself so don't know precisely how cycle times correlate with walking times!), and even a somewhat fiddly one involving a short walk/cycle!
All of these options can be paid for in their entirety with Oyster or contactless payment cards, or by use of a paper London Travelcard.
Mill Hill Broadway
This is probably the simplest route in terms of catching the trains; it also requires the longest cycle, however. At Paddington, follow the signs to the Hammersmith & City Line. Note there are two pairs of Circle Line platforms at Paddington at opposite ends of the mainline station; you want the ones that are also Hammersmith & City Line platforms, not the ones that are also District Line platforms. To get to the Hammersmith & City Line the best route is to take the staircase from the middle of any of the platforms up to the bridge over all the platforms (assuming you can take your bike up stairs; a few platforms also have a ramp if you're lucky. If you can't, you'll have to walk the long way round via the concourse to reach the lifts).
From the Hammersmith & City Line platforms at Paddington, catch an Eastbound train (Circle or H&C) to Farringdon. At Farringdon, change to the Northbound Thameslink platform; there are lifts to help you do this if need be. Note you could also change at King's Cross St Pancras but this station is much harder to negotiate than Farringdon so going one stop further will probably actually save you time.
From Farringdon catch a northbound stopping Thameslink train. Have a look at the station list, and ensure Mill Hill Broadway is in it (the train will likely have a final destination of Luton or St Albans). There seems to be a train which stops there about every 5-15 minutes (at uneven intervals) off-peak. Non-folding cycles are allowed on Thameslink trains subject to their policy — in short, no bikes on trains towards London timed to stop at Central London stations 07:00-10:00, or on trains away from London timed to stop at Central London stations 16:00-19:00 (since this isn't defined, I would personally assume Central London extends as far north as St Pancras International and no further; and as far south as London Bridge but no further, but that's just me).
Alight at Mill Hill Broadway. From there, cycle to Woodside Park; it's probably about 5-6km. This mostly follows the route of London Cycle Network Route 89 which can be seen on OpenStreetMap — though bear in mind that this will likely be quite a low quality cycle route, in that it will likely just be bicycles painted on quieter roads across London and not a dedicated segregated route or even lane in most places.
Cost: £4.60 (off-peak adult with Oyster or contactless)
Another option from Mill Hill Broadway is to cycle to Mill Hill East (including the option of cycling along part of the trackbed where the line used to continue further beyond Mill Hill East, which has now been converted into a cycle path), then catch the Northern Line train to Finchley Central, then another Northern Line train to Woodside Park. There's only one platform at Mill Hill East so that's quite straightforward, then changing at Finchley Central you should be able to just cross the platform island you arrive on to get to the next northbound train (no need to cross the line). This is obviously much more fiddly as it involves not one but two Northern Line trains, but it does involve a shorter and likely more pleasant cycle. The additional trains will cost an extra £1.50 on Oyster or contactless.
This is probably the middle route in terms of fiddliness compared with distance. Follow the Mill Hill Broadway instructions to get to Farringdon. From there, catch a Northbound Thameslink train that calls at Finsbury Park; the destination will likely be Cambridge or Peterborough. These run about every 15 minutes off-peak. You have the alternative option here of going to King's Cross St. Pancras as then you have more trains to choose from (some trains for Finsbury Park also run from King's Cross mainline station which don't pass through Farringdon), but again this is going to be much more fiddly so you'll probably have a smoother journey going the extra stop to Farringdon.
At Finsbury Park, alight the train, and board a Great Northern train which stops at New Southgate. These also run approximately every 15 minutes, and likely have a destination of Welwyn Garden City; do check that the train you're getting on stops at New Southgate! If you're curious, these trains have come from Moorgate, but you're not allowed to take cycles on trains between Moorgate and Drayton Green (one stop south of Finsbury Park) because this part of the line is basically an overgrown deep-level tube line! The Great Northern cycle policy for your purposes also restricts bicycles on trains arriving London 07:00-09:30, and leaving London 16:00-19:00.
From New Southgate you can cycle, again largely on Route 89, directly to Woodside Park. The distance is about 3.5km.
Cost: £4.60 (off-peak adult with Oyster or contactless)
The third option which stands out to me on reading a map is to go to New Barnet. This involves the shortest walk/cycle of the three, but is also the most fiddly. Follow the instructions above to get to Finsbury Park and on a train towards Welwyn, but alight a couple of stops later at New Barnet rather than New Southgate.
From New Barnet is quite a short walk or cycle to High Barnet on the Northern Line. The distance is about 2km and largely follows London Cycle Network Route 88.
From High Barnet, which is a northern terminus of the Northern Line, take the first train advertised to depart (there should be signs telling you which one this is). Alight at Woodside Park, and you're there!
Total cost: £6.50 (off-peak adult with Oyster or contactless)