First of all:
In which time and region do you want to set your story? The First Age, Second Age, Third Age, and Fourth Age all have different scenarios and political situations; and the time about TA 1640 (in which the Great Northern Wars were fought) is vastly different from the time of T.A. 3018-3019 (the setting of the War of the Ring), or the early Fourth Age. Take a look at the timelines and the lands and realms of Arda and get familiar with their history to find out which kind of background suits you best.
After you have chosen time and region, try to get to know the lands and location of your setting, which places are relevant, which peoples and cultures dwell in the area, and which animals and monsters populate the region. Who are the lands' rulers and which powers are influencing them and their peoples? Is the Dark Lord active in the area? Are agents of the White Council active?
You can use locations or places described in source material or mentioned on maps. Think of what type or kind of location you wish to depict: is it a settlement (farm, manor, hamlet, village, town, city); a fortress (watchtower, castle, fort); or is it a forsaken place (ruin, cave, lost sanctuary, open place)? If you do not find maps, floorplans or descriptions in the source material you can simply look up for maps on the internet (for example if you wish to describe a Dunlending or Hillman village, you can search the web for maps of historical Celtic villages that you could use as inspiration).
What kind of groups are active in your adventure location? What people or tribe lives there? Who are the next neighbours and who are likely foreign visitors? Who are the local clans, families, houses, guilds? Are there local cults or religions? Who are local the citizens or nobles and leaders? What is the local economy like?
Often the sheer background will inspire an adventure plot.
The next steps:
Now that you have an idea about the background in which your story takes place, ask yourself: what kind of story do you want to tell? Is it for example again about another plot of the Dark Lord or one of his minions to overthrow the local free people, or is this a story which takes place among the normal populace of your region? If there is no mighty Warlord or dark cult behind it, what are the motivations of the antagonists? The protagonists? Even without the influence of higher powers there might be greed, ambitions, honor, love, jealousy, competition, hunger for knowledge, curiosity, laziness, stubbornness, piety, fanaticism, boredom and many other things involved or factors and motifs.
If you don't know how to start your story or how to develop a plot, do not hesitate to borrow from things you may stumble upon like books, comics, movies, tv series, old mythologies or daily news — almost anything can be used as an inspiration.
If you need to create new non-player-characters, you might wish to take a look at the races, cultures and peoples of Middle-earth again, maybe also take a look at professions and classes and find fitting names for your protagonists and supporting characters. Do you need creatures, monsters or animals, or plants or artifacts you might wish to built in?
Now that you have so far gotten a feel for the basic scenery, chosen antagonists, and developed a plot, you need to get the whole thing started. How do the heroes or protagonists get involved in the Story? Classic "hooks" include quest-giver NPCs contacting the Players (like Gandalf's classic line: "I'm looking for someone to share in an adventure!"), or maybe they get into trouble, stumble into a fight or quarrel, find a corpse or a curious artifact, suddenly find themselves target of a character or group out of unknown reasons. Or maybe the hero(es) are personally involved? Do they have any local connections or are otherwise known or acquainted with some local characters? What about their family or tribal/clannish or Guild/professional affiliations? If you lack any ideas, again do not hesitate to steal or borrow ideas from everything you can get your hands on to find a good starting point and motivation for your protagonists to get involved.