I have difficulty imagining the availability of only monthly data.  A monthly reading of a stream height or flow is not sufficient in most circumstances.  If you have only monthly data summaries, perhaps the raw hourly or daily data has been lost with time.  Monthly data with just rainfall and streamflow can help determine water balance as the remainder of rainfall minus streamflow is evapotranspiration and any deep seepage losses.  You might try dividing tims of year between dormant season and growing season.  There are various hydrologic tools, models, techniques to apply with limited data, such as SCS soil curve numbers of hydrologic response.  

A difficulty with monthly data may includ some months with high rainfall on last day, with most of the storm runoff from that storm occurring in next month.  So you may have a few months where rainfall and runoff totals appear unusual.

If you have any other stream and rainfall gauges in the same physiographic vicinity, even at some distance, with greater detail of readings, you might try to correlate.  Hydrologists learn tools in school, but also learn from reading hydrologic studies of catchment experiments and in consultation with other hydrologists.   

Without knowing more about your needs or intent, it is difficult to suggests the options.  I am concerned though that one reading a month for streamflow is sufficient for anything unless flow has almost no variation.

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In modeling, you will be constrained to months that have both rainfall and streamflow. Monthly data is coarse for use to evaluate daily or weekly variation.  Assuming you are interested in seasonality, using monthly data, start with say 5 years of streamflow to calibrate and other say 5 years to validate. Once a model is validated and calibrated, you might try to use rainfall information in the past to predict streamflow (extrapolation error is expected). 

Some potential models that are very simple you can try are:

TR55- https://www.nrcs.usda.gov/wps/portal/nrcs/detailfull/national/water/?cid=stelprdb1042901

Basinsim: http://www.vims.edu/research/departments/bio/programs/semp/basinsim/about.php

You can use time series analysis to derived a rainfall-runoff function with lags.

Again, more information is needed on the purpose of your study. There will be limitations in using only monthly data and it might work for some studies at that aggregate scale.