Drink them pure: By drinking your basic tea types pure – black, green, oolongs, whites and others - you will learn to associate taste and liquid color with those teas. Once this is achieved, the fun really starts as you can now delve into how different origins, ecosystems and processing techniques can diverge the basic flavour profile of these teas. 
Beware of decaffeinated teas: A note of caution if your goal is to reduce caffeine intake. Unfortunately, the decaffeination process ruins the tea flavour profile. Decaf teas are only a pale cousin of the original when it comes to flavours. Make sure you choose teas with low caffeine levels such as whites and greens, or lightly oxidised oolongs when aiming to train your palate and seeking to reduce caffeine intake at the same time. You can also choose to prefer a herbal (non caffeinated) tea altogether.
Drink them loose: Freshness in tea matters. Preparing the teas yourself from loose teas will allow you to smell the leaves, and the leaves will open up more fully compared to even the most posh, silky pyramid tea bag. This will heighten your tea senses and your overall tea experience. Looking at the tea will help you to develop your roadmap (below). Brewing a full-unbroken leaf should give you a more complex and rounded flavour profile than a broken leaf.
Pay attention to water: To brew high quality teas, use filtered, freshly boiled water and steep for the indicated time and temperature. Chlorinated water can dominate and reduce your ability to taste the teas true flavour profile. 
Sequence and pair correctly: Imagine you are going to an Asian fusion restaurant and you order a wonton soup and a spicy curry. Which one would you taste first? You would most likely start with the lightest dish. The same principle applies to teas. If you are going through several teas, start with the lightest (usually white, followed by green) and work your way down towards the darker teas like Oolongs and eventually Black. 
Have a roadmap: When you are starting out, it’s helpful to prepare your mind and taste buds for what you are seeking. Start at a high level. What would I expect from a white, green, oolong or black tea? Its great to have a roadmap. For this, it is good to consult the tasting notes or flavour description of teas and/or the tea flavour wheels available on the web (such as the one pictured by Camellia Sinensis) These flavor wheels are similar to those available for wine and coffee and help us appreciate the full spectrum of notes available in our teas.
Keep notes: This is both for fun as well as for training. Keeping notes will force you to name and describe the aroma, liquor, physical appearance and flavour (actual taste) of each tea. It may surprise you that you might not record the same details a week or so later, but what you observe may vary by the day as well as what you tasted before and after.