How can Americans visit Europe with as much flexibility and convenience as possible?

There’s several approaches you can take. They all involve some planning, and there are some extra marks of caution.

• Mix Schengen and non-schengen stays. The actual Schengen rules say for any given day in the Schengen area, you must have spent less than 90 out of the 180 preceeding days in the Schengen area. The EU even provides a calculator to help with this. So, you can make journeys to other places a US citizen can visit in between Schengen stays. The UK and Ireland are popular with Americans. The former Yugoslav states and Bulgaria and Romanina are usually quite accepting also. I think the visa-free stays are shorter, but you can also go to North Africa (although there are security concerns in some parts of some of these destinations).

• Take advantage of pre-Schengen bilateral agreements between the US and European nations. These can be used to extend your stay in that country beyond the 90 days. Note that you cannot use these as part of a "swap" Stratergy as you could with visits to the UK, as you remain within Schengen, thus accumulating days against the 90/180 counter making it illegal for you to go back to non-bilateral Schengen nations, even while your stay remains legal.

• Investigate if you have any claims to EU citizenship. The rules vary from EU nation to EU nation, but if either of you have say, grandparents from an EU nation, you may be able to claim citizenship there. This would allow you to stay in the EU indefintley (provided you could support yourself), and make remaining in non-EU Schengen states easier. It would also remove the issue (discussed further below) that you would not be able to (legally) operate your business while in Europe (well, for EU members. You may still have to stop working when in Switzerland/Norway).

• Look in to a variety of long-stay "D" type Schengen visas. These allow you to stay for the duration in the country that awards them (different countries have different rules), and only apply the 90-in-180 for journeys to other parts of the Schengen zone.

• Look in to actually emigrating, as that seems more like what you want to do. (Certainly your language is suggesting that). If you are fairly wealthy, this is often not too difficult (depending on just how wealthy, you might even be able to "buy" your way to citizenship with a large enough investment.). If you can get a residence permit, then things look similar to how they do on the long-stay visas mentioned above.

Important Note

As has been mentioned above in this answer, and in other answers, you need to be careful that you are funding this visit through savings from your remote business, and not continuing to run it. Exactly how much trouble you get in to depends on if you are caught, and which country/border agent by, but note that working (including remote working), is no permitted by the visa-free entry that US citizens receive to the Schengen area, nor to several neighbouring states. See this question about a couple who tried to run their remote business in the UK (non Schengen, but operates similar rules, although the UK does seem especially strict on the working restrictions).

I’ll give you some of my personal experience as a white American citizen living in France on a long-stay working Visa. Standard disclaimers apply, and this isn’t legal advice.

Firstly, I wouldn’t worry too much about border checks coming into France, and not at all while traveling in the Schengen area after you have already entered the country. Border checks are largely perfunctory, they ask minimal questions, and it is very common to come in on a one-way ticket. To my knowledge, there is no requirement to have concrete plans to return home, nor to have accommodations booked for the duration of your trip, if you are coming in as a tourist (i.e., no visa). Once you are in, you will be able to freely travel within the Schengen area. If controlled, you simply need to show your passport and entry stamp. Note that, while my experience has been entirely smooth, if you have darker skin or an Arabic sounding name, you may face additional scrutiny (just yesterday I walked past a group of French cops in a train station with a giant backpack on. They proceeded to search the pockets of the black guy who was walking–without luggage–directly behind me). That shouldn’t scare you off, as you still should not be denied entry just for having a one-way ticket, but do make sure all your ducks are in a row.

If you want to stay in the Schengen zone for longer than 3 months, you can apply for a long-stay visa in the United States before you leave. For a visitor visa, they are good for a year, can be renewed, and you simply will need to show the ability to support yourself financially for the duration of the trip. There are no hard and fast guidelines, but I think around €1500 per person per month is usually acceptable. Note that operating your business by phone during the trip almost certainly counts as working, and thus is illegal on this type of visa. It also will not help in terms of proving financial support. There are likely visas available to individuals who want to independently operate a business while in France. If you want to go that route, I would check the expat site, an attorney, and the local French consulate.

For renting an apartment, you will have no issues. 1-2 months is a bit short, so you will probably have to find a place specialising in short-term rentals, but otherwise you should be fine. Many cities have aparthotels for just this purpose. When I arrived, I stayed 2 months in an aparthotel with a full kitchen for not much more than a regular rental. I was even able to bring my dog. France also has quite a few rentals available on sites like Air B&B. A regular apartment rental (generally 3 years in France, with the option to leave with 3-months notice) will be harder; you may need a French cell phone, bank account, and a guarantor to make that happen.

If you plan to “operate” your business over the phone, you might actually be working in Europe. That’s not what a tourist does. There could be tax consequences, both home and abroad.

If you can refrain from working on your business on your holiday, options might be:

• Apply for D visa for France or Italy, then only time in other countries counts against the 90 days rule. If you want to spend several months sampling French wine and culture, those 90 days for the rest of Schengen might be enough.
• Come for 90 days, then travel to a Schengen country with pre-Schengen bilateral agreements with the US (e.g. Poland) or a non-Schengen country.