The visa, in one form or another, is a major cost in some business trips both in financial terms as well as the physical time required to get around to various Embassies.
Requirements differ from country to country and of course, you can only have one passport in one Embassy at any given time.
In an age of Globalisation, it is all to easy to forget these entry requirements. For British passport holders, this may be harder to remember than for holders of other country's passports - others may be used to having to get a visa for everywhere; British passport holders are more used to being able to going pretty much anywhere, without too much of a fuss.
There are two main types of entry requirement (I use this term "entry requirement" as some requirements are not true visas). Firstly there is the hard and fast visa and secondly, the "entry requirement" such as the ESTA required by the USA or the ETA required by the likes of Australia and (soon) Canada.
I want to make it clear that the responsibility for having a visa (and correct travel documentation) rests with the traveler and no one else. You turn up at an airport without the proper documentation you will (probably) be denied boarding, without compensation. Turn up in a country without the correct documentation and you risk arrest. A travel agent will check if a visa is required and an agent can point you in the right direction for getting one and, indeed, arrange one. You can (double) check requirements yourself on the IATA website here. The IATA site will give other health advice about health requirements (and for some countries these, too, are an entry requirement. The IATA list (as it states) is not definitive - but it is pretty close.
To get your visa, I suggest that you use one of the Consular Visa Services, such as Ross Consular or for Eastern Europe, The Russia House. Others are of course, available. Although a fee is charged (and sometimes quite a high fee) this is more than compensated for in terms of the time that can be needed in some instances and for the consular agent's knowledge of what paperwork will be required as well as making sure an application is complete.
As regards countries such as the USA and Australia, the pre-clearance is made online and a (small) charge is usually made. What these countries are doing, is not so much "granting a visa" as establishing that you are a person of good character - there is nothing that country does not like which is lurking in your past. If you fail the pre-clearance, that does not always mean you cannot travel, it means that you have to apply for a full visa - and that will take time.
Make sure that you are using the official website of that country. Some websites pretend to be the official website, in order to lure you into extra charges. Here is the official site for the USA and for here for Australia
For the regular traveler to many visa-prone destinations, then you can apply to have duplicate passports. This is also helpful if you travel between Israel and Arab countries. If you do hold duplicate passports, then be careful to make sure the correct passport details are included in any air travel booking.