I took over a very similar team. Nothing whatsoever in the minors, maybe one or two passable major leaguers, and everyone else had either aged tremendously or walked. The previous owner the team in a bad situation.
I basically just blew up and started over. My first season I went 71-91 with mostly the players I had and a few patchwork FA's. Second was 61-101 and third was 56-106. Some people see this as tanking but no one in my world ever accused me of it-- my previous team was terrible and I was legitimately building to seriously contend in the near future. I'd say it's truly "tanking" if you're terrible for more than 2-3 years, because if you do that, it basically means you're just hoarding great IFAs and draft picks for years so you can unleash them all at once, rather than steadily replenishing your system and trying to get back to relevance as quickly as you responsibly can. I've seen lots of teams string together 4 or 5 100-loss seasons (or more) and they do so just to pick up high draft picks. I just did it because I didn't want to sign any great players to long and expensive contracts while my team didn't have a chance at the playoffs. I was also a newbie so some of that was due to me not really understanding ratings my first season or so and accidentally putting a lot of my player money into guys who weren't very good.
During those seasons, I just drafted smart. I visualized what I wanted my team to look like in 3-5 years. I made the mistake of not caring too much about defense (something I've since rectified) but I made sure to pick up middle-of-the-order bats and good pitching. I didn't underestimate the value of relievers either and made sure I got some of them. I was very patient with them, and instead of rushing them to the majors, I made sure they all spent several years in the minor leagues. Better to be mediocre for a few years and be a contender after that than to be a .500 team right now but never improve because you rushed your prospects.
My fourth season I went 80-82 as a started to transition my minor leaguers to the majors. Next season I went 84-78 (breaking my own AL record, and the league record, for most wins by a last-place team. I happened to join an incredibly stacked division by pure chance). I think that 84 wins was artificially low as well-- my pythag says I should have won 89 that year, so I'd count the pythag as better evidence of steady improvement. During this time I signed a few free agents, namely pitchers to shore up my staff. My hitters were mostly taken care of with my prospects at this point. I'd say a great time to start splurging on legit FAs is when your prospects are coming up and your team is at the point where one or two free agents may be enough to put you over the edge.
This year I'm 88-53 with 21 games to play, and I still have guys in my minors that are poised to (hopefully) improve me further next year. My pythag also says I should be 91-50 so, again, my improvement is even more pronounced than my record shows. My only FA signing was last year's AL Cy Young winner, which helped greatly after my prized homegrown pitcher suffered a devestating injury. I shouldn't really call him a FA signing, as actually someone else managed to beat me in a bidding war, but I dealt my best pitching prospect to get him. It ran counter to what I'd been doing with my team before, but at the point that you're a contender, it becomes about winning (because you have to make winning your immediate goal eventually).
Aside from drafting smart, international IFAs can be great, but the best way to get the most out of them is to beat the market. Lots of times, a team will spent an entire IFA budget on one player, and another team will come in and pay bargain prices for 2 or 3 players who are comparable to that player but just came out later in the season. Don't get me wrong-- paying top dollar for a superstar IFA is a great idea if done correctly, but my proudest signings are probably the ones where I got a cornerstone player for between $9-$12m. Luck into those signings, and your team may be good very fast.
Edit: Oh yeah, almost forgot: never skimp on coaching. Without good coaches, your prospect will never improve and you'll always suck. However, a good trick is to splurge on coaching at the levels containing your best prospects and not caring about the others. So if it's your first or second season with the team and all your good prospects are in A ball or Rookie ball, no need wasting perfectly good prospect money on coaches for your AA and AAA squads that year if they'll just be instructing organizational filler.