Roxanne,
Your family were apparently Methodists. Methodism took a lot longer to become established in Ireland as a separate denomination than in England. In Ireland there was considerable resistance to separating from the Church of Ireland. It was 1815 before Methodists started to conduct their own baptisms. However because of continuing loyalty and other factors, many continued to use the Church of Ireland for sacraments for years after this and it was 1871 before all Methodists routinely performed their own baptisms.
For marriages, the earliest ceremonies conducted by a Methodist Minister in Ireland that I am aware of, date from 1835 (Belfast Donegall Square, the first Methodist church in Ireland). However in the mid 1800s there were only a few Methodist Ministers in Ireland (Methodism relied heavily on lay preachers). So the shortage of Ministers contributed to the continuing practice of marrying in the Church of Ireland. In addition, in the early years, many Methodist Meeting Houses were not licensed for marriages so that too contributed to couples marrying in the Church of Ireland.
So to summarise, you are unlikely to find many Methodist baptisms before 1820. Few marriages before the 1840s and only a handful for many years after that. If there are no Methodist records in the location you are interested in, I would search the Church of Ireland instead, as that’s the most likely place to find the relevant event.
If Michael and Letitia were born c 1809, then it’s safe to say they were not baptised as Methodists. Most likely they’d be baptised into the Church of Ireland. If they married around 1838 then it’s possible that they had a Methodist ceremony, but only 1 or 2 meeting houses offered that then. Statistically they are more likely to have married in the Church of Ireland. Also remember that tradition was to marry in the bride’s church after which she’d attend her husband’s. So Letitia’s denomination at birth is probably crucial to locating where they married.
I don’t think any Methodist records are on-line. Some Church of Ireland records are on the rootsireland site, include those for St. Anne’s the main parish church in Belfast. However there were a number of other C of I churches in the city and their records are not on-line. I can see about 3 C of I churches in Belfast with records for the 1830s and 4 Methodist meeting houses. The churches usually hold the originals of these records but there are also copies of most of the records in PRONI, the public record office, in Belfast. (A few are still held by the individual churches, particularly Methodist records). A personal visit is required to access the copies at PRONI. Access to the records there is free. This link explains what records exist, parish by parish:
https://www.nidirect.gov.uk/publications/proni-guide-church-records
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net