Coralie,
Statutory registration of births didn?t start till 1864 in Ireland and so you won?t find birth certificates for the children if they were born in 1863 or earlier. Prior to that you are reliant on church records for baptisms. To do that you need to have an idea of which church. And many of the records are not on-line. (There are copies of most surviving church records in PRONI in Belfast but a personal visit is required to look them up).
Marriages (save for RC) were registered in the statutory records from April 1845 onwards. I searched for a marriage between John Woods & Jane Gordon in the Strabane area 1845 ? 1865 but could not see it. Possibly they married further afield (both are fairly common names) or before April 1845.
Looking for deaths 1864 ? 1874, I saw only one registered in Strabane for John Woods that might fit. He was born c 1823 and died in 1872. For Jane Woods I saw two. 1841 ? 1871 and 1834 ? 1874 (the second is probably more likely given the age). Don?t know if you are aware of those deaths. If not, you might want to investigate. You can order photocopies from GRO Roscommon for ?4 or you can view them on-line on the GRONI website. (You have to sign up and buy credits).
For the childrens? baptisms, I would search the records for the following Presbyterian churches in the Strabane area:
Leckpatrick (Baptisms start 1838, marriages 1845)
1st Strabane (Baptisms start 1828, marriages 1845)
2nd Strabane (Baptisms start 1844, marriages 1864).
None of the above records are on-line anywhere so far as I am aware, and so you?ll need to get someone to look them up in PRONI for you.
Elwyn