WARNING: Level gains are permanent. There are many items which reduce the progress bar in a given stat, but they can do nothing to actual gained levels in a stat. As an example, a mag which has gained 5 ability levels will always have at least 5 ability, regardless of how many ability-lowering items are used. This means you must lower a stat before its bar completely fills to avoid gaining levels in that stat.
PURE OFFENSE: For pure offensive mags, the general strategy is to use weapons as your primary source of attack increase, with secondary items used to lower ability. Below are detailed specifics regarding each type of pure offensive mag. I have attempted to find a good mix of high leveling speed, good item availability, and low cost for these methods. The values below are close approximations which assume you exclusively use weapons found in the field and secondary items purchased from shops. Actual costs may vary.
PURE S-ATK: Pure striking mags are the most difficult pure mags to raise, but it is still relatively inexpensive and straightforward to do so. Use hunter weapons to increase Satk. Because monomates are an inefficient way to lower ability and 4* furniture is costly, you will instead rely on a mix of 2* and 5* furniture, purchased at the furniture shop for 300 and 400 meseta respectively. These will lower ability rapidly, and if used alternatingly, the remaining bonuses will be canceled out. Your mag will need to consume 3 pairs of 2* and 5* furniture for every 180 S-atk points (6 pre-50 levels) gained. This will cost 350 meseta per 30 S-atk points, with an eventual total cost of 211,167 meseta for a lvl 150 S-atk mag.
PURE R-ATK: Pure ranged mags are arguably the easiest pure mags to raise. Use ranger weapons to raise Ratk. Rely solely on 5* furniture items (400 meseta each) to lower ability. Your mag will need to consume 6 5* furniture items per 180 R-atk points (6 pre-50 levels) gained from weapons; in doing so, your mag will gain an additional 30 R-atk points (1 pre-50 level) from the furniture itself. This will cost roughly 343 meseta per 30 R-atk points, with an eventual total cost of 206,857 meseta for a lvl 150 R-atk mag.
PURE T-ATK: Thanks to recent price decreases, pure tech mags are now the cheapest pure mag to raise. Use force weapons to raise T-atk. You will use either Shiftarides, Debanrides, or both (200 meseta each) to lower ability. Your mag will need to consume 15 rides for every 360 food points (12 pre-50 levels) of T-atk gained from weapons; in doing so, your mag will gain an additional 60 T-atk points (2 pre-50 levels) from the rides. This will cost roughly 214 meseta per 30 T-atk points, with an eventual total cost of 129,286 meseta for a lvl 150 T-atk mag.
PURE DEFENSE (SDEF, RDEF, or TDEF): I've seen very little demand for pure defensive mags, but I'm sure somebody will try this eventually. In this case, your strategy will be nearly identical to that mentioned for the equivalent attack stat, with the following modifications. For Sdef, substitute Leg units for Hunter weapons. For Rdef, substitute Arm units for Ranger weapons and 2* furniture for 5* furniture. For T-def, substitute Rear units for Force weapons.
PURE ABILITY: That leaves the pure ability mag. If you're going this route, you're lucky, as pure ability mags are fairly straightforward to raise. Ability mags like to eat two things: sol atomizers and discs. These items only raise ability and no other stats, making feeding straightforward and simple. Discs are significantly faster, and if possible, its diet should be wholly constituted by them.
ABILITY HYBRIDS: I've noticed a significant chunk of players who enjoy mags which evenly mix ability and one offensive stat. This strategy has a significant advantage in leveling speed because of the factors detailed in the Growth Slowdown section. Also, such mags can very easily be tuned to adopt a wide variety of final evolutions, with minimal investment in secondary stats. If you wish to build an ability hybrid, I recommend you primarily feed your mag the appropriate type of weapon, supplementing its diet with discs to cause ability to catch up. I have one note of serious caution for those who pursue this route. Decide early whether you want a 2nd evolution mag based off ability or off your chosen offensive stat. Then be very careful to make sure that your chosen stat is higher when your mag hits lvl 95, as its evolutionary form at that level will determine which mag family it belongs to for the remainder of its life. See the Mag Evolution section for more details.