If you want to test out the waters retinol is fine (wouldn't go for the Ordinary personally, Indeed Labs Retinol Reface is a good affordable starter retinol, Medik8 is good if you have more money to spend and want a brand that really specialises in good retinoid formulas, the Crystal Retinal is very popular) but agree with My3Cents in that you're basically building up to tretinoin, you don't necessarily have to and can just get on the Tret train from the start if you prefer (provided you don't have sensitive/reactive skin or a skin condition like rosacea). Not everyone is comfortable with that and through a subscription service like Dermatica or Skin and Me it is a cost of £20 a month (unless you space it out a bit).
Whichever you start with (go most conservatively with tretinoin) make sure you have a SPF50 that you like and a price to match, a 50ml bottle should last you about a month otherwise you're not getting the necessary protection, it's not something you can scrimp on in a little bit of product goes a long way. So add that to your monthly cost as well, SPF in moisturiser is not adequate but potentially you could skip your moisturiser in the AM if your SPF is moisturising enough.
Also useful to have some sort of barrier repair cream at hand like La Roche Posay Cicaplast Baume b5.
Start with using a retinoid once or twice a week at night and very slowly build up use to usually every other day (depends on the person). Avoid the area around the eyes and corners of mouth and inner corners of the nose liberally or place a barrier cream on it before applying your retinoid.
You can also buffer the retinoid if you're cautious by applying moisturiser on first then following with the retinoid cream.
Purging can happen but it should be brief, especially with the prescription stuff expect skin to go through a period of being not so great (dryness, flakiness, suddenly being a bit hairier) before it gets better.
If you're using any other actives, exfoliators, etc. dial down or stop using those altogether in the build up phase of introducing the retinoid. When using those again, introduce them gradually.