It's not finished yet, so it's still pretty empty. But I can't stop worrying about the rivers and mountains, so I ask for help. It's about 1100-1200 miles from north to south.
I notice that there are two parallel mountain chains in the upper left corner. This would be a natural place for a river in between them, parallel to both, running to the sea in the west. Instead, the upper chain does have a river running to the east.
That's not unheard of, but I'm wondering why the eastern side of the valley between the mountains would be at a lower elevation than the western side, as this implies. Maybe, since the mountain chains get closer together toward the west, there are hills connecting them, similar to foothills around the large mountain range on the right side of the map?
Edit Ok, so here's the slightly-more-informed opinion.
Since you have three north-south mountain ranges that will affect where your rain shadows are, I went to The True Size and compared India to the Western US.
Then I went to Google's My Maps and sketched out a trapezoid about the same shape as your landmass. Since the subtropical band starts at 25 degrees north, that's where I put the base of the trapezoid, just above the word Baja.
So to begin with, you've got a J-shaped crescent of subtropical weather along your southern and eastern coasts. At 25 degrees, the southern coast is like Florida, and the very top of the J, in the upper-right hand corner, is going to be like Virginia. Whether or not it's swampy coastal plain like the East Coast depends on soil deposition. But you might want to consider that, on the East Coast, a mountain range with a similar area as the vertical oval would have a dozen unique watersheds to the sea - and that's without being on the windward side of a peninsula.
Since you've got the wind coming from the east instead of the west, what you've basically created between these two mountain chains is a flipped version of the Western US. The area on the western sides of your mountains, opposite the direction of the wind, will be deserts and plains caused by rain shadows. Putting the southern shore at 25 degrees makes the landmass southerly enough that half the rain shadow will be a hot desert (in red), and half will be a temperate desert. But there isn't going to be much precipitation to create the snowmelt and runoff necessary to sustain more than one major river from that side, and it will probably be in the upper, more temperate half.
As your air masses continue moving west, they'll hit the second mountain chain. If it's high enough compared to the first one, then there will be enough snowpack to sustain at least three, maybe four major tributaries that join the river running south. (If your civilization has the means to build irrigation canals out of the river, and depending on soil quality, that could be some pretty productive farmland.) Since it's flipped, the western side of the Rocky Mountains is the kind of climate and precipitation pattern I'm thinking of here.
Finally in this sequence, you're going to have a dangerous, foggy western coast as the dry, hot air falling down the mountains mixes with the cold, wet air over the ocean.
Since there's a significant break between the eastern mountain chains, it allows for a large continental climate pattern in the upper part of your map.
And last but not least, the range in the upper right hand corner looks large enough, in conjunction with how the other northern chains are oriented, to create a large cold desert in the north, similar to Mongolia. This entire area will probably only have one major river, although there could a tributary from each individual range. If your range in the upper left hand corner is high enough relative to the right-hand-corner-range, then it can create a humid continental microclimate where that river would most likely start. In that case, the rivers from these mountains look about right (assuming you directed the river in that direction with foothills or something in the western side of that valley.)
Hope I didn't send you too far back to the drawing board.
The things that struck me are, first of all, the more-or-less random placement of mountains. Mountains have causes, and they can come from subduction of a oceanic plate (e.g., the Andes) or from a hotspot (e.g, the area around Yellowstone or Hawaii), or from plate collisions (e.g, the Alps and Himalayas) etc. There will be some patterns which are not visible here.
(I notice that all the mountains appear to be young. Is this intentional, or just the limitations of the drawing?)
Secondly, the land is drawn as flat with narrow mountain chains. You might, possibly, get one narrow mountain chain, but in general they will have lands to one or both sides which are rugged, but not mountains and there will be ranges of hills that don't make it to the status of mountains. I see none of this here.
I like the rivers! They start in mountains and they move plausibly across the land. (They don't flow uphill or do other impossible things like split and go off it two different directions.)
Rivers almost always flow away from mountain sources on both sides of a mountain range, not just one, as is the case in several of the mountain ranges shown, unless one side is a desert (which doesn't obviously seem to be the case, if it is, a lighter yellow or white color should be used for deserts). Indeed, even if there is a desert, it is common to have a river bed that is seasonally dry or even a narrow water fed green strand running through what is otherwise a desert, sometimes in a canyon and sometimes not (a la the Tigress and Eurphrates in Syria and Iraq, or the Nile in Egypt, Sudan and Ethiopia, or the Grand Canyon in the American Southwest).
Also, when a mountain range is the source of multiple rivers (as the Rocky Mountains in Colorado are), the sources of all of the major river systems tends to be centrally located near the point of highest elevation in the mountain range, and they tend to be close to each other. You could visit the source of all seven of the major river basins that originate in Colorado (the Colorado f.k.a. the Grand, the Arkansas, the Rio Grande, The N. Platte, the S. Platte, etc.) in two or three days by horse. Wyoming and Montana, Utah and Nevada, and New Mexico, to the north, west and south of Colorado include mountains in the same Rocky Mountain range as well, but are not the source of any really major river basins (they do have sources for some moderate sized tributaries that merge into the basins of bigger rivers eventually).
Similarly, almost all of the major river systems of mainland Southeast Asia and Eastern India have origins that are quite geographically close to each other in a central, high elevation area.
The main exceptions to this principle are mountain ranges that run along a coast. They tend to have lots of very fast running short rivers that run to the ocean or sea originating all along the mountain range.
Another unusual feature is the complete absence of lakes. Even in places where there are no really big lakes, rivers typically have some places that are much wider than the rest of the river which are called lakes and look like lakes on a map.
A Lack Of Endorheic Basins
Another thing that your map lacks, which isn't wrong, but is an opportunity missed, is the lack of an endorheic basin, which is to say a water basin that drains to an inland lake or sea, rather than ultimately to an ocean.
In the real world, endorheic basins almost always develop linguistically and culturally distinctive cultures that go undisturbed by the larger cultural forces of their region that obliterate other relic cultures, because humans almost always explore new territory by tracing rivers from the coast to their sources and rarely take the final step of crossing the water basin divide across the highlands that surround an endorheic basin until much later.
One notable issue with the mountains is that they superficially all seem to be about equal in height. But, Nature is not a big fan of equality. In the real world, mountain ranges tend to be unequal in height, some greater and some lesser, and usually have variation in height even within the same chain.
Mountain chains also typically have a few mountain passes midway, often not terribly straight, that allow them to be traversed somewhere at lower altitudes. Mountain passes and the entrances to them on both sides of the mountain range, tend to be culturally important trade cross-roads kinds of places that are also focal points for invading and defending armies on either side of the range.
Mountains also tend to form in chains separated by places of low elevation if there are parallel chains. If you have two mountain ranges running side by side closely enough to be part of the same geological system, you'd expect either a Great Basin (a la the Western U.S. between the Sierra Nevada range and the Rocky Mountains) or a chain of lakes (a la the Rift Valley of Africa), between them, not a big flat plain, which is something that develops as far from the mountains as possible. Sometimes the lowlands are submerged below water (the fjords of Scandinavia or the sea between Baja Mexico and the rest of Mexico).
The mountains in the map also seem to universally end rather abruptly. Real mountain ranges tend to have some intermediate foothills between the peaks and the plains unless they are isolated volcanos formed by migrating hot spots like the Hawaiian islands.
The mountain range in the Southeast of the map, in particular, looks wrong. If all of the other mountain ranges near the coast hug the coastline, so should this one.
Don't Be Too Hard On Yourself
The criticisms in this answer are rather blunt. But, you can take it or leave it. Nobody ever stopped reading a story or book because the geology isn't realistic by Earth standards (well, maybe a geology nerd or two might, but that's 0.01% of your audience, tops). The map looks pretty and fits a lot of genre conventions that date back to Tolkien at least. As long as it does the job, it doesn't matter if it is scientifically plausible.
But, if you want to add that extra polish and verisimilitude, you could revise it along some or all of the lines suggested in this answer.
It also doesn't have to be an all or nothing thing. Even a couple of relatively minor changes along the lines of some of these suggestions will make the map as a whole seem more realistic.
For example, simply widening one or two of the rivers into lakes in a couple of places and moving the Southeast mountain range closer to the coast would make the entire map seem more real, even if you change nothing else.
That actually looks extremely good for a fantasy map, which are drawn different than an actual landform map, the rivers make sense even if the origins on rivers are never a single point but I assume the map only notes impassable rivers. The mountains are not ill placed although that could change if you include elevation, the western ranges should be from the same cause thus should be connected by highlands. I assume the prevailing winds blow east to west in which case your northern portions may be too green.
I'd day it seems like it is pretty low in forest but it could easily be becasue most if it is settled farmland and you said it was not finished.
Depending on if your trying to depict climate it will be off, but since you are just asking about mountains and rivers they will work.