This is a broad question, but in simple terms it is answerable, in giving brief summary of some of the main items you should consider:
- Methods of Investing
- Tax Considerations
- Currency
- Administration
1) Methods of investing
The primary keyword you should be searching for is "International Savings Plan". There may be other terms used, but this is what I am familiar with, and it is geared to people in your situation, with uncertain tax status and multiple possible future countries of contribution and withdrawal. Be careful of accepting 'sales pitch material' at face-value, but firms that offer these funds should be familiar enough with your situation to be able to offer some guidance; as with anything involving technical complexity you are not familiar with, make sure you get multiple opinions before choosing a provider, and be extremely cautious to consider whether the provider you choose is legitimate before sending them money.
You will find options for these plans somewhat similar to a 'standard' pension plan offered in many countries, meaning you would be able to contribute money and allocate it to a set number of funds. Selection of those funds would be too broad for a single question, but as a starting point, look for low-fee, broadly diversified funds that are heavily weighted to equities, at least while you are young and able to benefit from the risk / reward associated. You can search other questions on this site about investing for further discussion on this.
2) Tax considerations
Unfortunately, you are unlikely to be able to significantly benefit from available options on tax deferral usually offered. There are 3 tax elements to consider:
a) Is your contribution tax-deductible in your current year?
This may be possible, but likely only if you open a new plan each time you move countries. International plans typically aren't registered with local governments in the way necessary to achieve immediate tax advantage. After moving, you may need to 'give up' this tax benefit, possibly even to the extent of paying tax on funds you have invested, to the extent you received an initial deduction.
b) Is the income earned within your plan deferred from paying tax until the future, instead of being taxed every year like a regular investment?
This again may be possible depending on how you structure things, but quite likely as soon as you change countries, passive income earned within your plan might be taxed in your new country of residence. The complexity of having multiple plans in multiple countries is also likely to be quite a headache for tax filings, and also quite expensive to hand off to an accountant, as complexity increases. There are simplistic methods available to you to attempt to defer some of the tax - primarily, investing in growth funds that don't pay dividends, meaning you may be able to significantly defer tax until you recognize capital gains on selling for future retirement withdrawals. {Even in that case, you may need to pay taxes on the interim growth of your savings at the point you cease to be a tax resident of any given country}.
c) Are the withdrawals during retirement going to be double-taxed in your future country of residence + original country of contribution?
This may be a significant concern, most especially of you are a US citizen [meaning you will file US taxes until you renounce citizenship, regardless of where you live]. If you are not a US citizen, you might end up paying double-tax depending on the relationship between the various countries involved, and their relevant tax treaties.
In order to know whether your contributions would (a) provide immediate tax relief; (b) allow tax deferral on future income; and (c) penalize you with double-taxes on withdrawal, you would need to know what countries are involved. No certainty is possible for generalized situations, and unfortunately the most likely answer regardless is 'assume 0 tax benefits, and a tax headache every year'.
3) Currency
If you know where you will retire, then it is your retirement currency which matters most. Investing with EUR funds would give you some security that if you retire in the Eurozone, you would be hedged against the risk of currency fluctuation. If you don't know where you will retire, this is an incredibly hard question to answer. Investing in GBP but eventually retiring in Australia would mean that you would need to convert your GBP to AUD on retirement, and if the GBP is weakened relative to the AUD in 30 years, that could be devastating for you. You can search this site for other currency-related questions that touch on this.
4) Administration
As you mention, simplicity will be key here. Attempting to be purely maximal in terms of tax advantages, currency considerations, and yield of return, may turn this into a monthly nightmare for you. I strongly agree with you that in order to build the habit of saving and begin to take advantage of the benefits of long-term compounding, you should attempt to select a straightforward plan that minimizes your need to take action every paycheque. This would be the main advantage of seeking a formal 'international savings plan' instead of attempting to open a local account every time you move. This will come at a financial cost, but at least to start with, is quite likely to be worth it.
One way to consider this - if the complexity builds so high that you aren't comfortable starting to invest for the next 3 years, then you have lost 3 years of saving, returns, and compounding, and that is quite likely to impact you negatively down the road.
Most importantly - set your calendar every year, likely the month before filing your taxes, to perform a complete re-investigation of your financial and life goals. Today, you don't know where you will live the next decade, or where you will retire in 30 years. Yet, you should start to save and invest as soon as possible, even if the solution you find first isn't the 'best'. And next year, you may know more about your future life plans, and at a certain point, a simpler / better option may present itself [ie: in 7 years, if you end up settling down in a given country, all of this complexity goes away, and you may want to collapse your current investment fund(s) and move everything into a local plan to maximize the tax benefits and align the currency needs].
Financial planning is an element of life planning, and you cannot do the one without the other.