Use of a wrap account/platform allows smaller minimum investments.
Macquarie Investment Accumulator would be my pick - I use it myself and have recommended it to younger clients wanting to start out small the same as I did. Also allows for gearing with a margin loan through a range of lenders. Extensive investment menu, good reporting capabilities. 0.62% p.a. admin fees and access to wholesale managed funds (lower fees because they're not bundled into the platform). Minimum initial investment is $2,000 with a regular investment facility set up, or $5,000 lump sum. Minimum amount per fund is $1,000.
Another option is MLC Navigator. Very open investment menu, includes some funds I'd love to use in Macquarie but can't, but otherwise similar features to Macquarie - minimum admin fee is $20 per month, so on smaller balances it can add up. A lot of positives though, but due to the high minimum fee really only superior if you're going to be regularly investing a reasonable amount. No minimum investment amount per fund but need $3,000 initial investment and a regular investment plan to open an account.
So with $2,000 to $3,000 you can establish a regular investment plan to get ongoing contributions automatically invested. If you wanted to, a lot of margin lenders will allow a relatively small initial loan drawdown so that you can ramp up the portfolio as your regular investments allow.
As far as minimum timeframes, unless you're looking at property trusts, mortgage income funds or other funds that are likely to be illiquid or restricted, they're not required timeframes but suggestions to not invest capital you'll need to access in the next 5-7 years. Volatility means you can't be sure you'll be above your purchase price if you needed to sell in the next few years.