Dave is right it does get easier. When I started there were no computers or even electronic calculators, so it was a question of using published tables plus a pencil and paper. These days computers make life easy and as Dave says you can check your answers using one. I remember checking to computer's answer against my long hand calculations when I first started using one
There are essentially two things you need to do:
Convert your local time of birth into GMT and then to Local Solar Time (that is time by the Sun not by the clock). LST is a very important step because Solar Time is continuous, whereas Time Zones suddenly change by an hour. For example, my birth time was BST, an hour ahead of GMT, at 01:30. That means my GMT birth time was 00:30. I was born at 2 degrees 11 minutes West of Greenwich. That means that when it's midnight in Greenwich, the LST for me is about 10 minutes earlier. This calculation is done using your longitude of birth. Then, using an ephemeris (I'd advise you to buy one, even though they can be found online) work out the zodiacal positions of the planets at your time of birth, expressed in GMT.
That involves calculating the fraction of daily travel that each planet has made since the previous midnight (the time given in the ephemeris) using the midnight positions on the day you were born and on the next day. So if you were born at the GMT equivalent of 6:00 a.m, then each planet will have moved 25% of it's daily travel and you can get the positions accurately for your birth. Dave mentions the acceleration of the interval but as long as you have the correct timings, that is a small adjustment and getting it wrong or missing it out is not a fail reason in itself (though if there are a series of errors it might contribute to it.)
Once you have the planetary positions for your time of birth then you need to plot those on your chart and this means finding your MC and ASC. For that you will need to have the table of houses and the longitude and latitude of the place. The table uses Sidereal Times, so you will need to convert your time of birth to ST. The ephemeris will give you the ST for each midnight and you can either do the calculation by hand or you can use a table which gives you the adjustments you need to make to the midnight ST to get your birth ST. What then follows is a look up in the table of houses and some interpolations to get the MC and ASC for your location.
Final step, draw out your chat wheel, mark the ASC and MC and then draw in the intermediate house cusps using your desired house system. Finish off by plotting the planets at their calculated positions..
Do ALL your charts between now and the exam that way, and keep the computer for presentation. Try doing one and posting your calculations here, just to keep your hand in and get some feedback. Look out for some books that cover the calculation step by step. I used to use Parker's Astrology, or Nick Campion's Practical Astrologer. but there are other books if you can find one still in print, such as John Filbey's Natal Charting.