While MJeffryes has provided a good card to help in the general case, I would like to answer your specific case.
There is no really good route, which can only be reached by train or metro. However, I can see some decent-looking routes when you're ready to take a longer walk or a short to medium bike (I'm not a cyclist myself, so I'm not sure how the bike times correlate with walking!), and even a bit awkward with a short walk / bike!
All these options can be paid in full with Oyster or contactless payment cards or with a London Travelcard in paper form.
This is probably the easiest route to reach the trains. however, it also requires the longest cycle. In Paddington, follow the signs to Hammersmith & City Line. Note that there is two Pairs of Circle Line platforms in Paddington at the opposite ends of the train station; They want those who are also Hammersmith & City Line platforms, Not those who are also district line platforms. To get to the Hammersmith & City Line, it is best to take the stairs from the middle of any platform to the bridge over all the platforms (assuming you can climb a staircase with your bike, some platforms also have a ramp, though) You're in luck, if you can not, you'll have to travel the long way across the hall to get to the lifts.)
From Paderston Hammersmith & City Line platforms, take a train eastbound (Circle or H & C) to Farringdon. In Farringdon, switch to the Northbound Thameslink platform. There are elevators that will help you when needed, although I think the platforms should be next to each other if my mental map is right. Note that you can also change at King's Cross St. Pancras, but this is the station much More difficult to negotiate than Farringdon. So if you go one step further, you'll probably save time.
From Farringdon take a northbound Thameslink train. Look at the station list and make sure Mill Hill Broadway is on the list (the train is likely to have a final destination in Luton or St. Albans). There seems to be a train that stops there about every 5-15 minutes (at odd intervals) outside peak times. Non-collapsible bicycles are permitted on Thameslink trains in accordance with their policy. In short, no bicycles in trains to London stopping at the stations in Central London between 07:00 and 10:00, or in trains outside of London stopping at Central London stations 16: 00-19: 00 ( since this is undefined, I would personally assume that Central London extends north to St. Pancras International and not further, and to London Bridge to the south, but no further, but that's just me).
Get off at Mill Hill Broadway. From there you cycle to Woodside Park; it is probably about 5-6km. This follows largely the route of the London Cycle Network Route 89, which can be seen on OpenStreetMap – but keep in mind that this is likely to be a low-quality cycling route, as it is probably only about bicycling on quieter roads London is painted and not In most areas there is a separate route or even a lane.
Another option from Mill Hill Broadway is to cycle to Mill Hill East (including the option of driving along a portion of the track bed that continued beyond Mill Hill East, which has now been converted into a cycle path) Northern Line – Train to Finchley Central, then another Northern Line train to Woodside Park. There is only one platform in Mill Hill East, so that's pretty easy. If you then change to Mill Hill Central, you should be able to cross the platform island where you arrive to get to the next train heading north (you do not have to cross the line). This is obviously much more complicated as it is not one but two Northern Line trains, but it is a shorter and probably more enjoyable cycle.
This is probably the middle route in terms of fumbling compared to the distance. Follow the instructions of Mill Hill Broadway to get to Farringdon. From there, take a Thameslink train northbound that stops at Finsbury Park. The destination will probably be Cambridge or Peterborough. These leave about every 15 minutes outside peak hours. Here you have the alternative option to drive to King's Cross St. Pancras, as you will have more trains to choose from (some trains for Finsbury Park also run from King's Cross, which is not through Farringdon but this is again the case to be a lot fumbler, you will probably have a quieter trip if you take the extra bus stop to Farringdon.
Get on the train at Finsbury Park and board a Great Northern train that stops at New Southgate. These also run approximately every 15 minutes and are likely to have a Welwyn Garden City destination; Make sure the train you board is in New Southgate! If you're curious, these trains came from Moorgate, but you're not allowed to take any trains between Moorgate and Drayton Green (one stop south of Finsbury Park), as this part of the route is basically deep subway! The Great Northern Bicycle Policy for your use also limits bicycles in trains arriving in London from 07:00 to 09:30 and departing London 16:00 to 19:00.
From New Southgate you can cycle directly to Woodside Park, also mostly on Route 89. The distance is about 3.5 km.
The third option that strikes me when reading a map is the drive to New Barnet. This includes the shortest path / cycle of the three, but is also the hardest. Follow the instructions above to get to Finsbury Park and take a train to Welwyn. However, get off a few stops later at New Barnet, not New Southgate.
It's just a short walk or bike ride from New Barnet to High Barnet on the Northern Line. The distance is approximately 2 km and follows mostly the London Cycle Network Route 88.
From High Barnet, a northern terminus on the Northern Line, take the first train that has been issued (there should be signs to tell you which one it is). Get off at New Southgate and you're here!
