The passage below is all we know of the Republic armed forces prior to the Clone Wars. The passage below occurs in 987 RE, whereas the Clone Wars began in 997 RE. (For an explanation of what RE means, look here.)
In moments, they had reached the shop that sold ruby bliels and had worked their way up to the counter.
Wald was as good as his word, and he produced the requisite druggats in exchange for three drinks and handed one to each of his friends. They took them outside, sipping at the gooey mixture through straws, and made their way slowly back down the street, chatting among themselves about racers and speeders and mainline ships, about battle cruisers and starfighters and the pilots who captained them. They would all be pilots one day, they promised each other, a vow they sealed with spit and hand slaps.
They were right in the middle of a heated discussion over the merits of starfighters, when a voice close to them said, "Give me the choice, I'd take a Z-95 Headhunter every time."
The boys turned as one. An old spacer stood leaning on a speeder hitch, watching them. They knew what he was right away from his clothing, weapons, and the small, worn fighter corps insignia he wore stitched to his tunic. It was a Republic insignia. You didn't see many of those on Tatooine.
"Saw you race today," the old spacer said to Anakin. He was tall and lean and corded, his face weatherworn and sun-browned, his eyes an odd color of gray, his hair cut short so that it bristled from his scalp, his smile ironic and warm. "What's your name?"
"Anakin Skywalker," Anakin told him uncertainly. "These are my friends, Kitster and Wald." The old spacer nodded wordlessly at the other two, keeping his eyes fixed on Anakin.
"You fly like your name, Anakin. You walk the sky like you own it. You show promise." He straightened and shifted his weight with practiced ease, glancing from one boy to the next. "You want to fly the big ships someday?"
All three boys nodded as one. The old spacer smiled. "There's nothing like it. Nothing. Flew all the big boys, once upon a time, when I was younger. Flew everything there was to fly, in and out of the corps. You recognize the insignia, boys?"
Again, they nodded, interested now, caught up in the wonder of corning face-to-face with a real pilot -- not just of Podracers, but of fighters and cruisers and mainline ships. "It was a long time ago," the spacer said, his voice suddenly distant. "I left the corps six years back. Too old. Time passes you by, leaves you to find something else to do with what's left of your life." He pursed his lips. "How're those ruby bliels? Still good? Haven't had one in years. Maybe now's a good time. You boys care to join me? Care to drink a ruby bliel with an old pilot of the Republic?"
He didn't have to ask twice. He took them back down the street to the shop they had just left and purchased a second drink for each of them and one for himself. They went back outside to a quiet spot off the plaza and stood sipping at the bliels and staring up at the sky. The light was gone, and stars were sprinkled all over the darkened firmament, a wash of silver specks nestled against the black.
"Flew all my life," the old spacer advised solemnly, eyes fixed on the sky. "Flew everywhere I could manage, and you know what? I couldn't get to a hundredth of what's out there. Couldn't get to a millionth. But it was fun trying. A whole lot of fun."
His gaze shifted to the boys again. "Flew a cruiser filled with Republic soldiers into Makem Te during its rebellion. That was a scary business. Flew Jedi Knights once upon a time, too."
"Jedi!" Kitster exhaled sharply. "Wow!"
"Really? You really flew Jedi?" Anakin pressed, eyes wide.
The spacer laughed at their wonder. "Cross my heart and call me bantha fodder if I'm lying. It was a long time ago, but I flew four of them to a place I'm not supposed to talk about even now. Told you. I've been everywhere a man can get to in one lifetime. Everywhere."
"I want to fly ships to those worlds one day," Anakin said softly.
Wald snorted doubtfully. "You're a slave, Annie. You can't go anywhere."
The old pilot looked down at Anakin. The boy couldn't look at him. "Well," he said softly, "in this life you're often born one thing and die another. You don't have to accept that what you're given when you come in is all you'll have when you leave."
He laughed suddenly. "Reminds me of something. I flew the Kessel Run once, long ago. Not many have done that and lived to tell about it. Lots told me I couldn't do it, told me not to bother trying, to give it up and go on to something else. But I wanted that experience, so I just went ahead and found a way to prove them wrong."
He looked down at Anakin. "Could be that's what you'll have to do, young Skywalker. I've seen how you handle a Podracer. You got the eyes for it, the feel. You're better than I was at twice your age." He nodded solemnly. "You want to fly the big ships, I think maybe you will."
He stared at the boy, and Anakin stared back. The old spacer smiled and nodded slowly. "Yep, Anakin Skywalker, I do think maybe one day you will."
This tells us very little about the Republic forces, but it does tell us a few things. We know the man was a pilot with the Republic's Fighter Corps, and we know he also manned the helm of larger vessels, "in and out of the corps". Presumably he was in the corps when he flew a cruiser filled with Republic soldiers to Makem Te, and also when he flew Jedi Knights. We may further speculate that the cruiser was itself a Republic vessel.
However, "cruiser" has been used in various ways in Star Wars. Leia's ship in ANH was called a "galactic cruiser" in chapter two of the novelization, but in chapter three it's called a "rebel craft" compared to Vader's "cruiser". The Super Star Destroyer is also described as one of "six cruisers" in the TESB novelization. Given that Leia's ship and the Super Star Destroyer are about a hundred times different in length, the term loses much of its potential meaning other than to refer to a ship larger than a fighter.
(Even the almost tiny Republic ship seen landing in the hangar deck of a Trade Federation battleship is called a "small Republic space cruiser", and the Nubian SR-71-esque vessel Amidala escapes from her planet in is called a cruiser by Trade Federation forces.)
Between the references above and the very notion of the "military creation act" from the other prequel films, it seems clear that the Republic's Fighter Corps (and/or its parent organization, since "Fighter Corps" seems rather limiting) prior to the Clone Wars was something roughly analogous to a UN peacekeeping force, perhaps with trappings of the Coast Guard. To be sure, the use of the term "battle cruiser" among the children may suggest the possibility of Republic vessels with more teeth, but t hen we don't know for sure what they were referring to. Leia's ship bristled with guns, after all.
But presumably, this Corp's personnel figured heavily in the full-fledged Republic Navy that fought in the Clone Wars, and it's entirely possible that the "Acclamator" Class vessels seen in Attack of the Clones were existing vessels of native Republic design, as opposed to Kamino-provided vessels.