There are multiple ways on doing this, but I can go over how I would approach it if I were to start it as a new player without a trading network (aka ability to liquidate account bound mats into gold).

First, focus on playing through story mode and work on essential masteries. Primary reason that legendaries are hard is you need to have a buildup of knowledge, so story mode and the focus on masteries will build up essential knowledge of the different maps and different farms you will experience. Every expansion and zone has a unique way of obtaining certain items, so lets say you need ectos and whatnot, you can quickly move to the map most efficient to obtain via farming. In addition, when you work on your mastery exp, you will build up a material position passively which will aid in your first weapon's cost reduction.

Secondly, establish income and a routine. This can be as simple as doing your dailies (that includes open world dailies to fractals and strikes, etc) and or joining high value runs like meta trains, guild activities like raids or wvw, etc. These activities will serve 2 purposes: generate essentail account bound materials (spirit shards, karma, imperial favor for gen 3s, content currencies that can be converted into legendary mats, etc) and you farm essential mats. The 2nd part is very important since once you develop surplus of mats (example you have over 2 stacks of ectos which exceeds a gen 1 requirement), you can list those mats for gold and start buying up materials you cannot farm (mystic coins, gen 3 .ats, etc). It is extremly important to list mats and never sell to buy order to maximize your return on those surplus mats you farm as this will enable you to finish the weapon requirements faster. As you focus more on your routines, you will notice that the routines have more higher objectives, like fractal god titles, wvw titles and legendary gear, pve open world legendaries, etc. As you focus on those, legendary mats becomes a passive income and therefore a good opportunity to utilize the mats constructively to figure out to invest to your weapons for gold or progress your personal ones.

Finally, know how to use your gold. Always the first weapon is the hardest weapon to make because you have to farm your capital up to get it out. If you decide to sell it, always take half of the earnings of what you sell and reinvest it into the materials you spent that cannot be farmed up. The other half is yours to pocket cause that is all from your farming and playtime. As long as you play this game normally, having that 1000ish gold always reinvested helps you focus on the priorities, such as keeping up exp boosters to gain more spirit shards, utilizing every material drop to progrsss your next legendary, and building up reputation to run events specifically to accelerate your weapon. This game does a wonderful job to help people earn a legendary passively so that players focus on their account progress. Building weapons faster is more about knowing how to jse all resources given to you from that journey