Every single unit in the game for each race has a specific strategical purpose and is specifically designed to be an effective counter against a specific unit(s) in the other two races. Indeed it is a game of specifics, and having a tight grasp of the underlying statistical mechanics at play is probably the most important thing for you to develop. As getting that attack upgrade means the difference between 2 shoting an enemy unit and 3 shoting an enemy unit (but of course you need to know 'which enemy unit' said unit is the best counter for in order to capitalize on the most efficient use of the attack upgrade as it doesn't class you a straight 1/3 cut in shoot to kill ratio across the board). In practice it is often the little details like these that can have the most drastic effect on the outcome of a given battle, which in turn may be the tipping point in your economy overpowering theirs (providing you have macroed a robust enough economy, which involves being competent enough at map control and multitasking to develop and safeguard your economy while scouting, teching, judging, reacting, microing e.t.c.), giving you the slight edge in unit production and ultimately winning you the game.
Understanding the symbiotic nature of starcraft goes a long way, as the mechanics are very detailed and very sensitive to each other, allot more-so than other RTS games, the more you play the more you will come to notice this. A beats B, D beats E, E beats C, C beats A, B beats C, D beats B, A beats E, C beats D,e.t.c.
On a practical basis, you need to get really good at doing 3 or 4 things at once, by 1) intellectualizing, thinking ahead, trying your best to judge your enemies every movement and possible strategy through map control and controlling the flow of information on the battle field. 2) physical play, by really getting used to the hotkeys and control groups and developing effective ways of being able to develop and manage your economy, tech development, creation and support of expansions and overall growth of presence on the map, while still being able to be 'elsewhere' on the map while doing all this, scouting, microing, battling, you name it, keep your background 'work' as 'background' and 'automatic' as possible and keep your eyes in the field and on the front line as much as possible, this is what i mean by multitasking, and this is serious multitasking.
Make sure you always try to Keep a clear and strategical mindset and develop enough of a knowledge base of each of the races to think three or four steps ahead, and act accordingly. Don't be purely re-active, be pro-active also. Come into a game with an idea of your strategical play for that match. Practice enough to really fine tune certain strategies you develop and develop a quick reflex to react to your opponent. Don't completely drop your strategy though, alter it, flow. Strategies should be fluid, They should be an ethereal impression of where you want to be, the tactics you deploy to get there are always a blank canvas. Luckily, due to the rigid mechanics of starcraft there is always a 'specific thing to do' in a given situation.
Basically, get to know the game, then get to know it even better, then make it your best friend, well... you get the point. Each race has a specific 'nature', learn these natures and learn how to counter them. For example the Zerg are known as the 'swarm' The basic phillosophy of zerg is aggressive expansion, map control and literally pouring your units at the enemy until they are over run. No other race can create units as fast as the Zerg, the 'creep' that follows the zerg wherever they go grants a 30% speed bonus to all zerg units. Think of the Flood from Halo. Mass insectoid face rape rampage style. The units are weak compared to the other races, so the idea is to outnumber the enemy and over run. This kind of stuff u need to find out. The nature of each race, and how those basic natures interact.
Then there are the unit counters. For example, again with zerg, zerglings are good against stalkers as stalkers are ranged units and are weak at close ranges, zerglings are also terrible against zealots as zealots are a 'meelee' unit slightly stronger than zerglings, Banelings on the other hand have a bonus Vs organics so rip through zealots, they are also brilliant against marines/marauders but terrible against any armoured type unit, like stalkers. Void rays are powerful against ground units but get ripped up by corruptors, which insidently can 'only' attack air units. Hydras are also good against void rays and other air units, they are ground based, and also very squishy with very low HP, if facing other ranged ground units, its an idea to keep them on the creep for the added speed to control the range and 'micro' them back and forth to keep them out of range of marines/marauders while still being able to shoot them from the edge of their range (unless your up against stalkers which i believe have a longer range than marines/marauders and of course blink, in which case lings would be a better strategy than hydras, again it depends on the attack/deff upgrades amongst a myriad of other situational elements). Or you could put a line of Roaches in front of your hydras, as roaches are the zergs mid game armored unit. Making great meat shields for the weaker hydras with a higher damage output and a longer range sat in back. Of course you could also use a pack of lings instead of you know your enemy is using ranged units for a more effective counter.
All of this requires you to have known 'this' circumstance in advance through controlling the flow of information, and to have built the required buildings to create the correct counter, manage your economy enough so you have enough supply formed and resources allocated theoretically to one side to be ready to create said counter force exactly when needed and be ready for this situation when it happens, instead of not knowing and letting them run you down with superior units that they have found out you have. Which all requires knowing what 'they are doing' almost always by quick glances at buildings they have, constant scouting and information gathering and strategical thinking in terms of where they might be expanding on the map or where they might be pushing on your ground.
Its a dance of minds. A strategical, emergent tango of minds... on crack.
I don't know if this has been of much help to you. I know there isn't much in the way of specifics. What i am trying to do is develop in you the right mind set. Help you get a feel for the underlying phillosophy, for the soul of this game, in its many beautiful ways.